Geometry

Types of Vectors: Zero, Unit, Parallel & More

The main types of vectors are the zero vector (magnitude 0), the unit vector (magnitude 1), and vectors classified by how they relate to one another: equal, negative, parallel, collinear, coplanar, and position vectors. This article defines each type, shows the diagram for it, and works through six examples of spotting the type from the numbers.

SAS Criterion in Triangles: Proof, Rules & Examples

The SAS criterion in triangles states that if two sides and the included angle of one triangle equal the two sides and included angle of another, the triangles are congruent. This article gives the rule, its transformation-based proof, worked examples, and the key reason SAS works while SSA does not.

Origins of Geometry: History from Egypt to Euclid

The origins of geometry lie in practical land measurement: Egyptians resurveyed flooded Nile fields and Babylonians tracked areas long before the Greeks turned those recipes into proof-based mathematics. This article traces geometry from Egypt and Babylon through Thales, Pythagoras, and Euclid, and explains why it is called "earth measurement."

Methods to Draw a Line Segment: Ruler & Compass

There are two methods to draw a line segment of a given length: the ruler method (mark both endpoints against a scale and join them) and the ruler-and-compass method (set the compass to the length, then transfer it as an arc). This article walks through both step by step with labelled figures, compares their accuracy, and works six examples plus the mistakes beginners make.

Geometric Constructions: Compass & Straightedge

Geometric constructions are exact figures drawn with only two tools - an unmarked straightedge and a compass - with no measuring. From five basic moves you can copy segments and angles, bisect them, drop perpendiculars, build parallels, and construct triangles, polygons, and circles. This hub lists every standard construction, explains the rules, and covers the three problems (trisecting an angle, doubling the cube, squaring the circle) proved impossible.

Euclid's Geometry: The Axiomatic System & 5 Postulates

Euclid's geometry is the plane geometry built in Euclid's Elements from a small set of definitions, five postulates, and common notions, with every later result proved by logic. This article gives an overview of that axiomatic system, states the five postulates in brief, and shows how the fifth led to non-Euclidean geometry.

Constructing an Angle of 60 Degrees: Compass Steps

Constructing an angle of 60 degrees with only a compass and straightedge means building one corner of an equilateral triangle, whose three angles are each 60°. Draw a base ray, sweep an arc from the endpoint, sweep an equal arc from where it crosses, and join the vertex to the intersection. This article shows the step-by-step image series, proves why it lands at exactly 60°, and works through six examples.

Components of a Vector: Resolving Into i, j, k

The components of a vector are the pieces it breaks into along each coordinate axis — the scalar components $x$, $y$, $z$ and the vector components $x\hat{i}$, $y\hat{j}$, $z\hat{k}$. Resolving a vector into components turns one slanted arrow into simple axis-aligned parts you can add and calculate with. This article covers 2D and 3D resolution, the formulas, and six worked examples.

Visualizing Solid Shapes: Nets, Views, and Euler's Formula

Visualizing solid shapes is the skill of picturing a three-dimensional object from flat information: unfolding it into a net, reading its top, front, and side views, and counting its faces, edges, and vertices - which always satisfy Euler's formula, $F - E + V = 2$. This article explains each tool with diagrams and works through six examples.

Triangle Inequality: Rule, Formula & Examples

The triangle inequality is the rule that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. For sides $a$, $b$, and $c$ this means $a + b > c$, $a + c > b$, and $b + c > a$ must all hold. A matching lower bound, the reverse triangle inequality, says the third side is longer than the difference of the other two: $|a - b| < c$. Together these give the range a side can take. This article explains the concept, why it is true, and how to test three lengths - the formal statement with proof lives in the triangle inequality theorem.

Tangents and Normals: Definition, Equations, and Examples

A tangent is a line that touches a curve at a single point with the same slope as the curve there, and a normal is the line through that point perpendicular to the tangent. Their slopes satisfy $m_{\text{tangent}} \cdot m_{\text{normal}} = -1$. This article covers both equations, their properties, the circle case, and six worked examples.

Spherical Coordinates: Definition, Formula & Conversion

Spherical coordinates locate a point in 3D space using three numbers: $r$, the distance from the origin, $\theta$, the angle down from the positive $z$-axis, and $\varphi$, the angle around the $z$-axis - written $(r, \theta, \varphi)$. This article defines the system, derives the conversion to and from Cartesian $(x, y, z)$, lists its properties, works six examples, and untangles the convention confusion over which angle is which.

Solid Shapes: Types, Properties & Formulas

Solid shapes are three-dimensional figures that occupy space - they have length, width, and height, and therefore volume. Every solid is described by its faces (flat or curved surfaces), edges (where two faces meet), and vertices (corner points where edges meet). Solids split into two families: polyhedra with only flat faces (cube, cuboid, prism, pyramid) and curved solids (sphere, cylinder, cone). This article defines solid shapes, lists their types and properties, gives the key volume and surface-area formulas with every variable named, and works six examples.

RHS Congruence Rule: Statement, Proof, Examples

The RHS congruence rule (Right angle, Hypotenuse, Side) states that two right triangles are congruent if the hypotenuse and one side of one equal the hypotenuse and the corresponding side of the other. This article states the rule, proves it from the Pythagoras theorem, separates it from SAS and SSA, and works through six examples plus the mistakes students make.

Rhomboid: Definition, Properties, and Formulas

A rhomboid is a parallelogram whose adjacent sides are unequal and whose angles are not right angles - so it is neither a rhombus (all sides equal) nor a rectangle (all angles 90°). Its opposite sides are equal and parallel, its opposite angles are equal, and its area is $A = b \times h$. This article defines the rhomboid, lists its properties, gives its formulas with every variable named, and works through six examples.

Representation of a Vector: Geometric and Algebraic Forms

The representation of a vector takes two main forms: a geometric arrow whose length is the magnitude and whose arrowhead gives the direction, and an algebraic form written in components $\langle x, y, z\rangle$, in unit-vector form $x\hat{i} + y\hat{j} + z\hat{k}$, or as a column matrix. This article covers every notation, defines each symbol, and works through six examples of converting between forms.

Properties of Parallel Lines: Angles & Rules

The core properties of parallel lines appear when a transversal cuts them: corresponding angles are equal, alternate interior and alternate exterior angles are equal, and co-interior (same-side) angles are supplementary, adding to 180°. Parallel lines also stay equidistant, share the same slope, and obey transitivity. This article states each property, proves the key ones, and works six examples plus the mistakes students make.

Polar Coordinates: Definition, Formula & Conversion

Polar coordinates describe a point in a plane by two numbers: $r$, the distance from a fixed origin (the pole), and $\theta$, the angle measured from the positive $x$-axis. A point is written as $(r, \theta)$ instead of the Cartesian $(x, y)$. You convert with $x = r\cos\theta$ and $y = r\sin\theta$ one way, and $r = \sqrt{x^2 + y^2}$, $\theta = \tan^{-1}(y/x)$ the other. This article defines the system, derives both conversion formulas, works six examples, and clears up why a point can have many polar names.

Plane Shapes: Definition, Types, and Examples

Plane shapes are flat, two-dimensional figures that have length and width but no thickness. This article defines plane shapes, lists their main types (triangle, square, rectangle, circle, and other polygons), explains their properties, sorts them into open and closed, and works through six examples plus the mistakes students make.

Perimeter of a Scalene Triangle: Formula & Examples

The perimeter of a scalene triangle is the total distance around it, found by adding its three unequal sides: P = a + b + c. This article covers the formula and variable key, the semi-perimeter shortcut, worked examples, the properties that make a scalene triangle special, and the mistakes to avoid.

Pentagram: Definition, Angles & the Golden Ratio

A pentagram is a five-pointed star formed by connecting every second vertex of a regular pentagon, written as the star polygon {5/2}. This article defines the pentagram, shows how to draw one, works out its $36°$ point angles, derives its hidden link to the golden ratio $\varphi \approx 1.618$, lists its properties, and separates it clearly from the pentagon.

Locus in Maths: Definition, Equation, and Examples

A locus is the set of all points, and only those points, that satisfy a given condition. This article defines a locus, shows how a circle, a perpendicular bisector, and a parabola each emerge from one rule, derives their equations from the distance formula, and works through six examples plus the mistakes students make.

Intersect in Math: Lines, Curves & Sets Explained

To intersect means to cross or overlap - to share a common point or element. Two lines intersect at a single point of intersection, found by solving their equations together. Two curves intersect wherever their equations are equal, found by substitution. Two sets intersect in the elements they share, written $A \cap B$. This article covers all three senses of "intersect," gives the point-of-intersection formula for two lines with every variable named, works six examples, and shows why some lines never meet.

Geometry in Daily Life: Uses & Examples

Geometry in daily life is the everyday use of shapes, angles, and spatial reasoning - from the hexagons in a honeycomb to the triangles in a bridge and the circles on every wheel. This article shows where geometry appears across nature, architecture, technology, sports, and art, and works through six real-world examples you can measure yourself.

Euclid's Fifth Postulate: The Parallel Postulate

Euclid's fifth postulate - the parallel postulate - says that if a line crossing two lines makes the interior angles on one side add to less than 180°, those two lines meet on that side. Its simpler equivalent (Playfair's axiom) is that through a point not on a line, exactly one parallel can be drawn. This article states the postulate precisely, explains why 2,000 years of attempted proofs failed, and shows how denying it created non-Euclidean geometry.

Euclid's Axioms and Postulates: The 7 Axioms and 5 Postulates

Euclid's axioms and postulates are the twelve self-evident assumptions on which all of classical geometry is built. The 7 axioms (Euclid's "common notions") are general truths about equality that apply across mathematics; the 5 postulates are assumptions specific to geometry, such as "a straight line can be drawn between any two points." This article lists all twelve, explains each, and clarifies how an axiom differs from a postulate.

Equation of a Plane: General, Normal & Intercept Forms

The equation of a plane in 3D space is $ax + by + cz + d = 0$, where $(a, b, c)$ is a vector normal (perpendicular) to the plane. This article covers the general, normal, point-normal, three-point, and intercept forms, derives each from a normal vector, works six examples including finding a plane through three points, and clears up the sign and normal-vector mistakes students make most.

Eccentricity of Parabola: Why It Equals 1

The eccentricity of a parabola is exactly $e = 1$, because every point on a parabola is the same distance from the focus as from the directrix. This article defines eccentricity, derives why a parabola gives $e = 1$, compares it with the other conics, and works through six examples.

Difference Between Cube and Cuboid

The difference between a cube and a cuboid is the shape and size of the faces: a cube has six equal square faces and all edges the same length, while a cuboid has rectangular faces and can have three different edge lengths. Both have 6 faces, 12 edges, and 8 vertices, so every cube is a special cuboid. This article compares their properties, formulas, and worked examples.

Construction of Angles : 30° to 120° With a Compass

The construction of angles uses only a compass and straightedge to draw exact angles without a protractor. A 60° angle comes from an equilateral-triangle arc, a 120° from two such arcs, a 90° from perpendicular arcs, and 30° and 45° come from bisecting 60° and 90°. This article gives step-by-step methods, the examples, and which angles are constructible.

Complete Angle : Definition, Measure, and Examples

A complete angle is an angle that measures exactly 360° - one full rotation of a ray about its endpoint, back to where it started. It is also called a full angle, round angle, or perigon, equals $2\pi$ radians, and is four right angles. This article explains how a complete angle forms, how it compares to other angle types, and works through six examples.

Collinearity : Conditions, Formula & Examples

Collinearity is the property of three or more points lying on one single straight line. Three points are collinear when any of these equivalent conditions holds: the slopes between them are equal, the area of the triangle they form is zero, the determinant of their coordinate matrix is zero, or the vectors joining them are parallel. This article explains what collinearity means, derives each test with its variables named, works six examples, and shows the mistakes that make students miss it.

Closed Shapes: Definition, Types & Examples

A closed shape is a figure whose boundary starts and ends at the same point, leaving no gaps - like a square, circle, or triangle. This article explains closed shapes, how they differ from open shapes, the main types, their properties, and where you see them every day.

Classification of Triangles: By Sides & By Angles

The classification of triangles sorts every triangle two ways at once: by its sides (equilateral, isosceles, scalene) and by its angles (acute, right, obtuse). This article explains both frameworks, shows how they combine into named types like a right scalene triangle, lists the properties that decide each class, works six examples, and clears up where students misclassify.

Basic Area Concepts: Unit Squares & Area of Shapes

Area is the amount of flat space a shape covers, measured in square units by counting how many unit squares fit inside it. This article defines area, shows the unit-square method, derives the area formulas for squares, rectangles, and triangles, works six examples, and clears up the mistakes students make when area and perimeter get tangled.

Area of Triangle in Determinant Form: Formula

The area of a triangle in determinant form is $\text{Area} = \frac{1}{2}\left| \det\begin{bmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{bmatrix} \right|$, where the three rows are the vertex coordinates. This article shows how to build and expand the $3\times 3$ determinant, why the absolute value and the $\frac{1}{2}$ appear, how a zero determinant proves three points are collinear, and six worked examples.

Arcs and Subtended Angles: Central & Inscribed

An arc subtends an angle wherever two lines drawn to its endpoints meet. The central angle (vertex at the centre) is always twice the inscribed angle (vertex on the circle) that stands on the same arc - the Central Angle Theorem. This article covers arcs, subtended angles, the theorem's proof, key properties, and six worked examples.

Angle Sum Property: Triangle, Quadrilateral, Polygon

The angle sum property states that the interior angles of a triangle always add to 180°, the interior angles of a quadrilateral add to 360°, and the interior angles of any polygon with $n$ sides add to $(n-2)\times 180°$. This article proves each rule from a single idea, gives the formula with every variable defined, and works through six examples plus the mistakes students make.

Analytical Geometry: Definition, Formulas, Examples

Analytical geometry (also called coordinate or Cartesian geometry) studies geometric figures by placing them on a coordinate system and describing them with algebraic equations. This article defines analytical geometry, gives the core formulas, the distance formula, midpoint, slope, and section, with every variable defined, and works through six examples plus the mistakes students make.

Alternate Segment Theorem: Statement, Proof, Examples

The alternate segment theorem states that the angle between a tangent to a circle and a chord drawn from the point of contact equals the inscribed angle that the chord subtends in the alternate segment. This article states the theorem precisely, proves it from the inscribed angle theorem, gives the converse, and works through six examples plus the mistakes students make.

X and Y Graph: How to Plot Points on the XY-Plane

An x and y graph is a flat grid built from a horizontal x-axis and a vertical y-axis, where every point is named by an ordered pair $(x, y)$. This article shows how to plot points, why the x-coordinate comes first, what the four quadrants mean, and works through six plotting examples.

Triangle Inequality Theorem: Rule and Examples

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is always greater than the third side: $a + b > c$, $b + c > a$, and $a + c > b$. This article explains the rule, proves it, shows how to test three lengths, and works six examples with the common errors to avoid.

Symmetry of a Circle - Lines and Rotational Symmetry

The symmetry of a circle is unlimited: a circle has infinitely many lines of symmetry - every line through the centre (each diameter) - and infinite rotational symmetry, because it looks the same after a turn of any angle about its centre. This article shows why, counts the lines, and clears up the common mix-ups.

Special Parallelograms: Rectangle, Rhombus, Square

The three special parallelograms are the rectangle, the rhombus, and the square. Each is a parallelogram with an extra condition: a rectangle adds four right angles, a rhombus adds four equal sides, and a square adds both. This article covers their shared properties, differences, and worked examples.

Rotational Symmetry - Order & Angle of Rotation

Rotational symmetry means a shape maps exactly onto itself when turned about its centre by some angle smaller than a full turn. The order counts how many times it matches in one $360°$ rotation, and the angle of rotation is $\dfrac{360°}{\text{order}}$; this guide gives the definitions, six worked examples, and the traps that produce a wrong order.

Perimeter of a Pentagon: Formula 5s, Examples

The perimeter of a pentagon is the total distance around its five sides. For a regular pentagon it is $P = 5s$, where $s$ is one side length; for an irregular pentagon you add all five sides, $P = a + b + c + d + e$. This guide derives both formulas, works six examples, and keeps perimeter cleanly separate from area.

Pentagonal Pyramid: Volume, Surface Area, Faces

A pentagonal pyramid is a 3D solid with a pentagon base and five triangular faces meeting at a single apex, giving 6 faces, 10 edges, and 6 vertices. Its volume is $V = \dfrac{1}{3} \times (\text{base area}) \times h$ and its surface area adds the base to the five triangles. This guide derives both formulas and works six examples.

Octahedron: Faces, Edges, Vertices & Properties

An octahedron is a polyhedron with 8 faces, and the regular octahedron is a Platonic solid built from 8 equilateral triangles with 12 edges and 6 vertices. This article defines it, shows its properties, its volume and surface-area formulas, how it obeys Euler's formula $F - E + V = 2$, and worked examples.

Negative of a Vector: Definition, Formula, Examples

The negative of a vector $\vec{a}$, written $-\vec{a}$, has the same magnitude as $\vec{a}$ but points in the exact opposite direction. You find it by multiplying every component by $-1$, and $\vec{a} + (-\vec{a}) = \vec{0}$. This article covers the formula, properties, six worked examples, and the mistakes students make.

Latus Rectum of Ellipse: Formula and Examples

The latus rectum of an ellipse is the chord through a focus, perpendicular to the major axis, and its length is $\frac{2b^2}{a}$ for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$. This article defines it, derives the formula, locates its endpoints, links it to eccentricity, and works six examples.

How to Find Slope From a Graph : (Rise Over Run)

To find the slope from a graph, pick two points the line passes through cleanly, then compute rise over run - the vertical change divided by the horizontal change. This article shows the step-by-step method, how to read the sign, worked examples, and the mistakes that flip an answer negative.

Hexahedron: Definition, Cube, and Examples

A hexahedron is any polyhedron with six faces, and the most famous one is the cube, the regular hexahedron, with 6 square faces, 12 edges, and 8 vertices. This article covers the definition, the cube's volume $a^3$ and surface area $6a^2$, the other types of hexahedra, and Euler's formula.

Hexagonal Pyramid: Definition, Volume, and Examples

A hexagonal pyramid is a 3D solid with a hexagonal base and six triangular faces meeting at an apex, giving 7 faces, 12 edges, and 7 vertices. This article covers its structure, the volume formula $V = \tfrac{1}{3} \times \text{base area} \times \text{height}$, surface area, worked examples, and the mistakes students make most.

Gradient Definition - Meaning & Formula in Maths

The gradient definition in mathematics is the measure of how steeply a line or graph rises or falls: it is the rate of change of $y$ with respect to $x$, calculated as $\dfrac{\Delta y}{\Delta x}$, and it is the same quantity as slope. This guide explains the meaning of gradient as steepness and as a rate, gives six worked examples, and clears up the mistakes that flip or misread the value.

Frustum of a Cone: Definition, Formulas, and Examples

A frustum of a cone is the solid that remains when the top of a cone is cut off by a plane parallel to its base, leaving two circular faces of radii $R$ and $r$. Its volume is $\frac{1}{3}\pi h(R^2 + Rr + r^2)$. This article covers the volume, surface-area, and slant-height formulas, with six worked examples.

Equilateral: Definition, Shapes, and Examples

Equilateral describes any shape whose sides are all equal in length, from the equilateral triangle to the rhombus and beyond. This article defines the term, separates it from regular and equiangular, shows how to find the perimeter of an equilateral shape, and works six examples.

Equation of a Line Parallel to the Y-Axis : (x = a)

The equation of a line parallel to the y-axis is $x = a$, where $a$ is the constant distance of the line from the y-axis. Every point on it shares the same x-coordinate, its slope is undefined, and this article covers the formula, six worked examples, and the common mistakes to avoid.

Equation of a Line Parallel to the X-Axis : (y = b)

The equation of a line parallel to the x-axis is $y = b$, where $b$ is the constant distance of the line from the x-axis. Every point on it shares the same y-coordinate, its slope is zero, and this article covers the formula, six worked examples, and the common mistakes to avoid.

Equation of a Line - All Forms, Formulas, and Examples

The equation of a line is an algebraic rule every point on a straight line obeys, and it can be written in six standard forms: slope-intercept $y = mx + b$, point-slope, two-point, intercept, normal, and general $Ax + By + C = 0$. This article shows each formula, when to use it, and how to convert between them.

Eccentricity of Hyperbola - Definition, Formula, and Examples

The eccentricity of a hyperbola measures how "open" the curve is, and it is always greater than 1. For the standard hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, it is given by $e = \dfrac{c}{a} = \sqrt{1 + \dfrac{b^2}{a^2}}$, where $c^2 = a^2 + b^2$. This guide derives the formula, contrasts it with the ellipse, and works six examples.

Directrix of Ellipse - Definition, Formula, and Examples

The directrix of an ellipse is a fixed straight line, perpendicular to the major axis, given by $x = \pm\dfrac{a}{e}$ for the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, where $e$ is the eccentricity. An ellipse has two directrices, one for each focus, and together with the focus they define the ellipse through the ratio $e < 1$. This guide derives the formula and works six examples.

Direction Cosine - Definition, Formula, and Examples

The direction cosines of a vector are the cosines of the three angles $\alpha, \beta, \gamma$ it makes with the positive $x$, $y$, and $z$ axes, written $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$. For a vector $a,\hat{i} + b,\hat{j} + c,\hat{k}$ they are each component over the magnitude, and they always satisfy $l^2 + m^2 + n^2 = 1$. This guide derives the formulas, proves that identity, and works six examples.

Diagonal of Hexagon - Number, Length, and Examples

A hexagon has 9 diagonals, obtained from the polygon rule $\dfrac{n(n-3)}{2}$ with $n = 6$. In a regular hexagon these split into 3 long diagonals through the centre (length $2s$) and 6 short diagonals (length $\sqrt{3},s$), where $s$ is the side. This guide derives the count, gives both lengths, works six examples, and clears up the double-counting mistake.

Definition of Patterns — Types, Rules, and Examples

The definition of patterns in math is a repeated or predictable arrangement of numbers, shapes, or objects that follows a rule. This article covers number patterns and shape patterns, the main types (arithmetic, geometric, Fibonacci, square, and triangular), how to find the rule, and the mistakes to avoid.

Cyclic Quadrilateral: Definition, Properties, and Examples

A cyclic quadrilateral is a four-sided figure whose four vertices all lie on a single circle, which forces each pair of opposite angles to add to 180°. This article defines it, proves the opposite-angle rule, lists every property, works through six examples, and names the mistakes that cost marks.

Convex Polyhedron: Definition, Euler's Formula, Examples

A convex polyhedron is a 3D solid with flat polygon faces where any line segment joining two surface points stays inside the solid. Its faces (F), edges (E), and vertices (V) always satisfy Euler's formula, $F - E + V = 2$.

Constructing Angle Bisectors — Definition, Steps, and Examples

Constructing an angle bisector means splitting an angle into two equal halves using only a compass and straightedge — no protractor. The method: draw an arc from the vertex cutting both rays, then two equal arcs from those cut points, and join the vertex to where they meet. This guide gives the exact steps, proves why they work, and shows the special case that builds a right angle.

Constructing a 90 Degrees Angle: Compass Steps

Constructing a 90 degrees angle with only a compass and straightedge means building a right angle from equal-radius arcs, no protractor needed. This article walks through the compass method step by step, shows why it produces exactly 90°, gives six worked constructions, and flags the errors that ruin the arc.

Concurrent Lines: Definition, Conditions, and Examples

Concurrent lines are three or more lines that all pass through a single common point, called the point of concurrency. This article defines them, shows the determinant condition for testing concurrency, walks through the four concurrency points of a triangle, and works six examples with the mistakes to avoid.

Concentric Circles: Definition, Formula, and Examples

Concentric circles are two or more circles that share the same centre but have different radii, so they never touch. This article defines them, gives the annulus area formula $\pi(R^2 - r^2)$, separates them from congruent circles, works six examples, and lists the mistakes to avoid.

Concave Shape: Definition, Reflex Angle, Examples

A concave shape is one that caves inward: it has at least one interior angle greater than 180° (a reflex angle), and you can draw a straight line between two inside points that passes outside the shape. This guide explains the reflex-angle test and the diagonal test, covers concave 2D and 3D shapes, and works six examples.

Circumscribe: Circle, Polygon, and Examples

To circumscribe means to draw one figure around another so it touches it. A circumscribed circle (circumcircle) passes through every vertex of a polygon; its centre is the circumcenter and its radius the circumradius. This article covers circumscribed circles and polygons, the formulas, and the difference from inscribed figures.

Centroid of a Triangle — Definition, Formula, and Examples

The centroid of a triangle is the point where its three medians intersect, and it is the triangle's balance point. Its coordinates are the average of the vertices, $G = \left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3}\right)$, and it divides every median in the ratio 2:1 from the vertex. This guide gives the formula, the ratio, and six worked examples.

Cartesian Coordinate System: Definition, Quadrants, and Examples

The Cartesian coordinate system pins down the exact position of any point using ordered pairs $(x, y)$ measured from two perpendicular axes that meet at the origin. This article defines the axes, the four quadrants and their sign rules, ordered pairs, how to plot points, and the 3D extension into octants.

Area of Similar Triangles: Ratio Theorem, Examples

For two similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides. If the sides are in ratio $k$ (the scale factor), the areas are in ratio $k^2$: $\dfrac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\dfrac{AB}{DE}\right)^2$. This guide states the theorem, proves it, and works six examples in both directions.

Axis of Symmetry of a Parabola - Definition, Formula, and Examples

The axis of symmetry of a parabola is the straight line that cuts the curve into two mirror-image halves, passing through the vertex, the focus, and perpendicular to the directrix. For a parabola $y = ax^2 + bx + c$, it is the vertical line $x = -\dfrac{b}{2a}$. This guide derives that formula, connects it to the vertex and focus, and works six examples.

Argand Plane - Definition, Modulus, and Examples

The Argand plane is a coordinate plane that pictures a complex number $z = x + iy$ as the single point $(x, y)$, using a horizontal real axis and a vertical imaginary axis. This article shows how to plot complex numbers, read off the modulus $|z| = \sqrt{x^2 + y^2}$ and argument, and how the Argand plane differs from an ordinary Cartesian plane.

Area of a Quadrant: Formula, Definition, Examples

The area of a quadrant is one-fourth of a full circle's area, so it equals $\dfrac{\pi r^2}{4}$, where $r$ is the radius. This article defines a quadrant, derives the formula from the full circle, covers the diameter form and the perimeter, and works six examples.

Angle Between Two Lines: Formula and Examples

The angle between two lines is found from their slopes with $\tan θ = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|$, which returns the acute angle at the intersection. This article covers the slope formula, the line-equation and 3D vector forms, the parallel and perpendicular special cases, and six worked examples.

Adjacent Angles of a Parallelogram: Definition & Rule

The adjacent angles of a parallelogram are the two angles that share a side, and they are always supplementary - they add up to $180°$. This article shows the rule in a figure, proves why it holds, works through six examples, and names the errors that trip students up.

Midsegment of a Triangle - Definition, Theorem, and Examples

The midsegment of a triangle is the segment joining the midpoints of two sides. By the midsegment theorem it is parallel to the third side and exactly half its length. This article defines the midsegment, proves the theorem, works through examples, and clarifies how the midsegment relates to (and differs from) the midpoint theorem.

Types of Quadrilaterals: Definition and Classification

A quadrilateral is any four-sided closed shape whose interior angles sum to $360°$, and the main types of quadrilaterals are the parallelogram, rectangle, square, rhombus, trapezoid, and kite. This article defines each type, shows how they nest in one family tree, and works through examples of classifying a shape from its sides, angles, and diagonals.

Vertex of an Ellipse: Definition, Formula & Examples

The vertices of an ellipse are the two endpoints of the major axis - the farthest points from the center - located a distance $a$ from the center along the longer axis. The endpoints of the shorter (minor) axis are the co-vertices, at distance $b$. This article defines both, gives the formulas for finding them from an ellipse's equation, and works through examples.

Vertex Of Hyperbola - Definition, Formula, and Examples

The vertices of a hyperbola are the two points where each branch is closest to the centre, lying on the transverse axis a distance $a$ from the centre. For $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ they sit at $(\pm a, 0)$, and the distance between them is $2a$. This article defines the vertex, derives its coordinates from any standard equation, and works through examples.

Sum Of Exterior Angles Of A Triangle — Why It Is Always 360°

The sum of the exterior angles of a triangle is always $360°$ — taking one exterior angle at each of the three vertices. This holds for every triangle, whether it is scalene, isosceles, equilateral, acute, right, or obtuse.

Rhombus Lines of Symmetry: 2 Diagonals Explained

A rhombus has exactly 2 lines of symmetry, and both of them run along its diagonals. It stops at 2 (rather than the square's 4) because a rhombus is a tilted diamond, so folding along a side-to-side midline never makes the halves match.

y = mx + c - Definition, Formula, and Examples

y = mx + c is the UK form of the straight-line equation, where m is the gradient (slope) and c is the y-intercept. This article defines each term, derives the equation, shows how to graph it and read $m$ and $c$ straight off it, and clears up the one thing that confuses everyone: $y = mx + c$ and $y = mx + b$ are the same equation in different notation.

Platonic Solids — Definition, Properties, and Examples

The platonic solids are the five convex 3D shapes whose faces are all identical regular polygons meeting the same way at every corner: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. This article proves why only five can exist, tabulates their faces, vertices, and edges, and shows how Euler's formula $F + V - E = 2$ checks each one.

Parallelepiped: Definition, Volume & Scalar Triple Product

A parallelepiped is a three-dimensional solid bounded by six parallelogram faces, with opposite faces parallel and congruent. Its volume equals the absolute value of the scalar triple product of the three edge vectors meeting at a corner: $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$. This article defines the solid, renders it with measurements, and works through the volume calculation step by step.

Oval Shape: Definition, Properties, Oval vs Ellipse

An oval is a smooth, closed, curved 2D shape that looks like the outline of an egg — no straight sides, no corners, no vertices. The word is informal: it names a family of egg-like curves rather than one exact shape.

Midpoint Theorem - Statement, Proof, and Examples

The midpoint theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and exactly half its length. This article gives the formal statement, a full labelled proof, the converse, six worked examples, and the common mistakes, plus how the theorem relates to the triangle's midsegment.

Lines Of Symmetry In A Parallelogram - Why The Answer Is Zero

A general parallelogram has 0 lines of symmetry - there is no straight line you can fold it along so that the two halves match. Its diagonals split it into equal areas but not into mirror images, and no midline works either.

Icosahedron - Definition, Properties, and Examples

An icosahedron is a three-dimensional solid with 20 equilateral-triangle faces, 30 edges, and 12 vertices, and it is one of the five Platonic solids. Its surface area is $A = 5\sqrt{3},a^2$ and its volume is $V = \dfrac{5(3+\sqrt{5})}{12},a^3$ for edge length $a$. This article defines it, counts its parts, derives both formulas, and works through examples.

Geometrical Constructions - Definition, Steps, and Examples

Geometrical constructions are precise drawings of lines, angles, and shapes made using only a compass and an unmarked straightedge - no measured lengths or protractor. This article walks through the core constructions: the perpendicular bisector, the angle bisector, a perpendicular from a point, and a 60° angle, each with numbered steps and the reasoning behind them.

Dodecahedron - Definition, Formulas, and Examples

A dodecahedron is a three-dimensional solid with 12 flat faces; the regular dodecahedron, one of the five Platonic solids, has 12 identical regular pentagon faces, 20 vertices, and 30 edges. This article gives its properties, its surface area $3\sqrt{25 + 10\sqrt{5}},a^2$ and volume $\frac{15 + 7\sqrt{5}}{4},a^3$ formulas, and six worked examples.

Distance Between Two Planes: Formula and Examples

The distance between two parallel planes $ax + by + cz + d_1 = 0$ and $ax + by + cz + d_2 = 0$ is $\dfrac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}}$, provided their normal coefficients match. This article shows why the planes must be parallel for a distance to exist, why non-parallel planes have distance zero, and works through six examples

Distance Between Two Lines: Formula and Examples

The distance between two lines is the length of the shortest segment joining them, measured perpendicular to both. For parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ it is $d = \dfrac{|c_2 - c_1|}{\sqrt{a^2 + b^2}}$; for skew lines in 3D it uses the cross product. This article defines both cases and works through examples.

Distance Between Point and Plane: Formula & Examples

The distance between a point $(x_0, y_0, z_0)$ and a plane $ax + by + cz + d = 0$ is the shortest, perpendicular distance, given by $\dfrac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$. This article derives the formula, explains what every term does, works six examples, and shows that a distance of zero means the point lies on the plane.

Area vs Surface Area: Difference, Formulas, Examples

The difference between area and surface area is dimensional: area measures the space inside a flat 2D shape, while surface area measures the total of every face on the outside of a 3D solid. Both are reported in square units; this article gives the definitions, formulas, worked examples, and the mistakes students make when the two are confused.

Cylindrical Coordinates - Definition, Formula, and Examples

Cylindrical coordinates locate a point in 3D space as the ordered triple (r, θ, z), where $(r, \theta)$ are the polar coordinates of the point's shadow on the xy-plane and z is the usual height. This article gives the conversion formulas to and from Cartesian coordinates, shows when the system is worth using, and works through six examples plus the mistakes to avoid.

Converse of Pythagoras Theorem - Proof and Examples

The converse of the Pythagoras theorem states that if the square of a triangle's longest side equals the sum of the squares of the other two - if a² + b² = c² - then the triangle is right-angled. This article gives the statement, a full proof, how the test also flags acute and obtuse triangles, and six worked examples plus common mistakes.

Concave Polygons: Definition, Properties, Examples

A concave polygon is a polygon with at least one reflex interior angle greater than 180°, which makes at least one vertex point inward and at least one diagonal fall outside the shape. This article defines concave polygons, contrasts them with convex ones, lists their properties, and works through examples like the arrowhead and the five-pointed star.

Compass Drawing - The Tool, Its Parts, And How To Use It

A drawing compass is a hinged, two-legged instrument used to draw circles and arcs: one leg holds a sharp needle that pins the centre, the other holds a pencil that swings around it. The gap between the two legs sets the radius.

Cartesian Form - Definition, Formula, and Examples

Cartesian form writes a point, line, or plane using the coordinates x, y, and z directly, with no parameter or direction vector attached. This article defines Cartesian form, gives the point, line, and plane formulas, contrasts it with vector and polar form, and walks through six worked examples plus the mistakes students make most.

Area of Triangle in Coordinate Geometry: Formula & Examples

The area of a triangle in coordinate geometry is found from its three vertices with the formula $\text{Area} = \frac{1}{2}\left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$, the shoelace formula. This article derives the formula, works through six examples, and shows how a result of zero proves the three points are collinear.

Area Of Polygons - Definition, Formula, and Examples

The area of a polygon is the flat space it encloses. A regular polygon uses $A = \tfrac{1}{2},P,a$ (half the perimeter times the apothem); an irregular polygon is found by splitting it into triangles or by the shoelace formula $A = \tfrac{1}{2}\left|\sum (x_i y_{i+1} - x_{i+1} y_i)\right|$. This article defines both routes and works through examples.

Annulus - Definition, Area Formula, and Examples

An annulus is the flat ring-shaped region between two concentric circles - circles sharing a centre but with different radii. Its area is $\pi(R^2 - r^2)$, where $R$ is the outer radius and $r$ the inner radius. This article defines the annulus, derives its area and perimeter, works through examples, and clears up the mistakes students make.

Angles of a Square - Interior, Diagonal, and Sum of Angles

Every interior angle of a square measures exactly 90°, so the four angles sum to 360°. Its two diagonals cross at the centre at 90° and split each 90° corner into two 45° angles. This article proves each of these facts, works through examples, and separates the square's angle behaviour from that of a general rectangle or quadrilateral.

Angles of a Rectangle - Properties and Examples

Every one of a rectangle's four interior angles is 90°, and they add to 360°. This article explains why all four are right angles, what the diagonals do to those corners (they split each 90° into two acute angles and are equal but not perpendicular), and works through six examples plus the mistakes students make.

Vector Form - Definition, Formula, and Examples

The vector form of a line writes every point on it as $\vec{r} = \vec{a} + \lambda\vec{b}$, so you start at a known point $\vec{a}$, then slide any distance $\lambda$ along a direction $\vec{b}$. It is the cleanest way to describe lines and planes in three dimensions, where the ordinary $y = mx + c$ equation breaks down.

Sum of Angles in a Polygon - Formula and Examples

The sum of the interior angles in a polygon equals $(n-2) \times 180°$, where $n$ is the number of sides. This article shows why that formula works (a polygon splits into $n-2$ triangles), works through pentagons, hexagons, and irregular shapes, and separates the interior-angle sum from the exterior-angle sum, which is always 360°.

Square Pyramid - Definition, Formula, and Examples

A square pyramid is a 3D solid with a square base and four triangular faces meeting at an apex, giving 5 faces, 8 edges, and 5 vertices. Its volume is $V = \tfrac{1}{3}a^2h$ and its surface area is $a^2 + 2a\ell$. This article covers the definition, properties, net, formulas, and worked examples.

Sphere: Volume, Surface Area, and Great Circle Explained

For a sphere of radius $r$, the volume is $\frac{4}{3}\pi r^3$ and the surface area is $4\pi r^2$ - exactly four times the area of its great circle ($\pi r^2$). Every flat slice through a sphere is a circle, and the biggest such slice, through the centre, is the great circle.

Rise Over Run - The Slope Method Explained

Rise over run is the step-by-step method for finding the slope of a line: count the vertical change (rise, $\Delta y$) and divide it by the horizontal change (run, $\Delta x$). It turns "how steep is this line?" into a fraction you can read straight off a graph or compute from two points. This guide gives the method, six worked examples, and the mistakes that flip the sign or the fraction.

Reflex Angle: Definition, Degree Range, and Examples

A reflex angle is any angle greater than $180°$ and less than $360°$ - the larger opening on the outside of an ordinary angle. This article defines the reflex angle, shows how to measure one with a standard $180°$ protractor using the $360° - \theta$ rule, explains why $180°$ and $360°$ are excluded, and works through examples from clock hands.

Reflection Symmetry - Definition, Mirror Line, and Examples

Reflection symmetry exists when a single straight line - the mirror line, also called the line of symmetry - divides a shape into two halves that are exact mirror images of each other. Every point on one side has a matching point the same distance away on the other side.

Protractor in Math - Parts and How to Measure Angles

A protractor is a semicircular tool that measures and draws angles in degrees, from 0° to 180°. This article explains its parts (centre, baseline, and the two scales), shows step by step how to measure and draw an angle, works six examples, and settles the biggest confusion: which of the two scales to read.

Position Vector - Definition, Formula, and Examples

A position vector is a vector that points from the origin to a given point, fixing that point's location in space. For a point $P(x, y)$, its position vector is $\vec{OP} = x\hat{i} + y\hat{j}$. This article covers the formula in 2D and 3D, how to find the vector joining two points, its magnitude, and how it differs from a displacement vector.

Pentagonal Prism - Definition, Formula, and Examples

A pentagonal prism is a prism whose two bases are pentagons - five-sided polygons - connected by five flat rectangular faces. Because the two ends are identical and parallel, and the sides are straight rectangles, the shape belongs to the family of prisms, the same family as the rectangular box and the triangular prism.

Parallel Vectors - Definition, Condition, and Examples

Two vectors are parallel when one is a scalar multiple of the other: $\vec{a} = k,\vec{b}$. Equivalently, their cross product is the zero vector. This article covers the definition, the scalar and cross-product conditions, same-direction versus anti-parallel cases, worked examples, and the mistakes to avoid.

Orthocenter - Definition, Properties, and Examples

The orthocenter is the single point where a triangle's three altitudes meet. It sits inside an acute triangle, on the right-angle vertex of a right triangle, and outside an obtuse triangle. This article shows how to find it by hand and on the coordinate plane, and how it differs from the circumcenter, incenter, and centroid.

Negative Slope - Meaning, Graph, and Examples

A negative slope means a line falls from left to right - as x increases, y decreases. It is found with $m = \dfrac{\Delta y}{\Delta x}$, and it comes out negative whenever the two variables move in opposite directions. This article covers what negative slope means, how to calculate it, its graph, real examples, and the common mistakes.

Latus Rectum: Parabola, Ellipse, and Hyperbola Formulas

The latus rectum is the focal chord of a conic drawn perpendicular to its main axis, ending on the curve; its length is $4a$ for a parabola $y^2 = 4ax$ and $\dfrac{2b^2}{a}$ for both an ellipse and a hyperbola. This article defines the latus rectum, gives each conic's formula and endpoint coordinates, explains why it measures a conic's width at the focus, and works through examples.

Inscribed Angle Theorem - Statement, Proof, and Examples

The inscribed angle theorem states that an angle inscribed in a circle is half the central angle that subtends the same arc — so $\angle$inscribed $= \tfrac{1}{2}\angle$central. This article gives the exact statement, proves it in three cases, derives its main corollaries (angle in a semicircle is $90°$; angles on the same arc are equal), and works through examples.

Hypotenuse - Definition, Formula, and Examples

The hypotenuse is the side of a right triangle that lies opposite the right angle, and it is always the longest side. You find it with the Pythagorean theorem, $c^2 = a^2 + b^2$; this article covers the definition, the formula, six worked examples, and the mistakes students make most.

Hexagonal Prism: Faces, Volume, and Surface Area

A hexagonal prism is a 3-D solid with two parallel hexagonal bases joined by six rectangles, giving 8 faces, 18 edges, and 12 vertices. This article defines the hexagonal prism, derives its volume $V = \tfrac{3\sqrt3}{2}a^2h$ and surface area $SA = 3\sqrt3,a^2 + 6ah$, shows its net, and works through examples, starting with the pencil in your hand.

Heptagon: 7-Sided Polygon, Angles, and Examples

A heptagon is a seven-sided polygon with 7 sides, 7 vertices, and 7 interior angles that add up to 900°. In a regular heptagon every side and angle is equal, so each interior angle measures about 128.57° and each exterior angle about 51.43°.

Gradient of a Line - Definition & Formula

The gradient of a line measures its steepness and direction: it is the change in $y$ divided by the change in $x$, and it is exactly the $m$ in the equation $y = mx + c$. "Gradient" is the British term for what is called slope elsewhere - they mean the same thing. This guide gives the formula, six examples, and the mistakes that flip or fumble the value.

Frustum - Definition, Volume, and Surface Area

A frustum is the 3D solid that remains when you slice the top off a cone or pyramid with a cut parallel to the base. For a conical frustum with radii $R$ and $r$ and height $h$, the volume is $\frac{1}{3}\pi h (R^2 + r^2 + Rr)$; this article gives the volume and surface-area formulas for both cone and pyramid frustums, with six worked examples.

Eccentricity - Definition, Formula, and Examples

Eccentricity (e) is a single number that tells you the shape of a conic section: e = 0 for a circle, 0 < e < 1** for an ellipse, **e = 1** for a parabola, and **e > 1 for a hyperbola. This article gives the formula for each conic, worked examples, where eccentricity matters, and the common mistakes.

Distance of a Point From a Line - Formula & Examples

The distance of a point from a line is the shortest length between them, measured along the perpendicular dropped from the point to the line. For a point $(x_0, y_0)$ and a line $Ax + By + C = 0$, that distance is $d = \dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$; this article derives the formula and works six examples.

Diagonal of Rectangle - Formula & Examples

The diagonal of a rectangle is found with $d = \sqrt{l^2 + w^2}$, where $l$ is the length and $w$ is the width - it is the hypotenuse of the right triangle each diagonal creates. This guide derives the formula, works six examples, and clears up the mistakes that give wrong diagonal lengths.

Decagon - Definition, Sides, Angles, Diagonals, and Area Formula

A decagon is a ten-sided polygon whose interior angles sum to $1440°$; in a regular decagon each interior angle is $144°$ and each exterior angle is $36°$. This article defines the decagon, derives its area formula $A = \tfrac{5}{2}a^2\sqrt{5+2\sqrt{5}}$, counts its 35 diagonals, and works through examples, starting with the ten-sided coin in circulation today

Corresponding Sides in Geometry - Definition, Examples

Corresponding sides are sides in the same relative position in two figures that are either similar or congruent. In congruent figures corresponding sides are equal; in similar figures they are proportional - their ratio is constant. This article shows how to match them from vertex order, how to use the ratio to find a missing length, and the mistakes to avoid

Convex Shape - Definition, Properties & Examples

A convex shape is a shape where the straight line joining any two points inside it stays completely inside - equivalently, every interior angle of a convex polygon is less than $180°$. This guide gives the one-line test, the properties, six worked examples, and the mistakes that make students misclassify a shape.

Angles of a Quadrilateral - Formula and Examples

The four interior angles of a quadrilateral always add up to 360°, because any four-sided figure splits into two triangles of 180° each. This article shows why the rule holds, how to find a missing angle, the exterior-angle and cyclic-quadrilateral cases, and the mistakes to avoid.

Acute Triangle - Definition, Properties, Examples

An acute triangle is a triangle in which all three interior angles are less than 90°. Its three angles still add to 180°, its orthocenter lies inside it, and it can be equilateral, isosceles, or scalene. This article covers the definition, properties, how to test a triangle from its sides, the area formula, and worked examples.

180 Degrees to Radians - Value and Conversion Steps

180 degrees is exactly $\pi$ radians - about 3.1416 radians. This guide gives the quick answer, a degree-to-radian reference table, where the half-turn shows up, the formula behind it, three methods, and the mistakes that trip students up

2D Shapes - Names, Properties, and Examples

2D shapes are flat figures with only two dimensions, length and width, and no thickness. This article names the common 2D shapes (circle, triangle, square, rectangle, pentagon, hexagon, octagon), explains their sides, vertices, and angles, shows how to find area and perimeter, and clears up the mistakes students make most

Zero Slope: Definition, Equation, Graph & Examples

A zero slope is the slope of a horizontal line, where the rise is 0, so the line is perfectly flat and its equation is y = c. This guide shows you how zero slope is calculated, how it looks on a graph, how it differs from an undefined slope, and the mistakes students make most often.

X and Y Intercept: Definition, Formula & Examples

The x-intercept is where a graph crosses the x-axis (set $y = 0$ and solve), and the y-intercept is where it crosses the y-axis (set $x = 0$ and solve). This guide defines both, gives the find-it method, derives them from line forms, and works through six examples including a real distance problem.

Types of Polygon: Classification with Examples

Polygons are classified along four independent axes: by number of sides (triangle, pentagon, hexagon), by regularity (regular vs irregular), by shape (convex vs concave), and by structure (simple vs complex). This article works through every type with properties, a comparison table, examples, and the classification mistakes to avoid.

Two Point Form: Formula, Derivation & Examples

The two point form gives the equation of a straight line from any two points on it, using the formula $y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)$. This guide derives the formula from slope, works through clean and tricky examples, explains when it cannot be used, and lists the common mistakes.

Triangles: Definition, Types, Properties, and Examples

A triangle is a closed two-dimensional shape with three sides, three angles, and three vertices, and its interior angles always add to $180°$. This guide covers a triangle's parts, the six types (by side and by angle), the core properties, area and perimeter formulas, six worked examples, and common mistakes.

Straight Angle — Definition, Properties, and Examples

A straight angle is an angle that measures exactly 180°, formed when its two arms point in opposite directions from a shared vertex and line up into a straight line. This article defines the straight angle, lists its properties, separates it from a straight line, and works through examples involving supplementary pairs and triangles.

SSA Congruence Rule: Why It Is Not Valid (Ambiguous Case)

The SSA congruence rule (two sides and a non-included angle) is not generally valid, because the same SSA data can produce zero, one, or two different triangles, the ambiguous case. This article explains why the SSA congruence rule fails, shows the two-triangle construction, and covers the one right-angle (RHS / HL) exception where it does work.

Square Prism: Volume, Surface Area Formulas, and Examples

A square prism is a 3D solid with two equal, parallel square bases joined by four rectangular faces. Its volume is $a^2 h$ (base side squared, times height), and its total surface area is $2a^2 + 4ah$ — the two square ends plus the four rectangular sides. This article derives both formulas, shows why a cube is just the special square prism where every face is a square, and works through examples and the mistakes students hit most.

Segment of a Circle: Definition, Area Formula, and Examples

A segment of a circle is the region between a chord and the arc it cuts off - the part of the disc sliced away by a straight line across it. This guide covers the minor and major segment, the area formula $\left(\frac{\theta}{360°}\pi r^2 - \frac{1}{2}r^2 \sin\theta\right)$ in degrees and radians, the perimeter, worked examples, and the mistakes to avoid.

Right Circular Cylinder: Definition, Volume, Surface Area, and Examples

A right circular cylinder is a 3D solid with two equal, parallel circular bases joined by a curved surface, where the axis joining the base centres stands perpendicular to the bases. Its volume is $\pi r^2 h$, its curved surface area is $2\pi r h$, and its total surface area is $2\pi r(h + r)$. This article derives each formula, explains exactly what makes a cylinder "right circular" rather than oblique, and works through examples and the slips students hit most

Right Circular Cone: Volume, Surface Area, and Examples

A right circular cone is a 3D solid with one circular base and a single apex that sits directly above the centre of that base, so its axis stands perpendicular to the base. Its volume is $\frac{1}{3}\pi r^2 h$, its curved surface area is $\pi r l$, and its total surface area is $\pi r(l + r)$, where $l$ is the slant height. This article derives each formula, explains exactly what makes a cone "right circular" rather than oblique, separates the height from the slant height, and works through examples and the slips students hit most

Pythagorean Triples: Definition, Formula, List, Examples

A Pythagorean triple is a set of three positive whole numbers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$, forming the sides of a right-angled triangle. This article covers Pythagorean triples in full: the definition, Euclid's formula for generating triples, primitive versus non-primitive triples, the common list, and the errors students hit most

Pyramid: Definition, Volume, Surface Area Formulas, and Examples

A pyramid is a 3D solid with a flat polygon base whose triangular faces rise to meet at a single point called the apex. Its volume is $\frac{1}{3} \times \text{base area} \times \text{height}$, and its total surface area is $\frac{1}{2} \times \text{base perimeter} \times \text{slant height} + \text{base area}$. This article derives both formulas, separates the height from the slant height (the most common confusion), and works through examples and the mistakes students hit most

Positive Slope: Definition, Graph & Worked Examples

A positive slope describes a line that rises from left to right: as x increases, y increases too, so the slope value $m$ is greater than 0. This guide shows you how to recognise a positive slope on a graph, how to calculate it with the rise-over-run formula, and the mistakes that trip students up.

Polar to Rectangular: Formula & Worked Examples

To convert polar to rectangular coordinates, use x = r cos θ and y = r sin θ, turning a distance-and-angle pair (r, θ) into an across-and-up pair (x, y). This guide derives the formulas from a right triangle, works through examples in every quadrant, and lists the common mistakes.

Polygons: Definition, Types, Properties, and Formulas

A polygon is a closed, flat shape made of three or more straight sides joined end to end, with no curves and no gaps. This article defines the polygon, classifies it by sides and by shape, derives the interior-angle sum formula $(n-2)\times 180°$, and works through examples and the mistakes students make most

Point of Concurrency in a Triangle - Types & Examples

A point of concurrency in a triangle is the single point where three of the triangle's special lines cross at once. This article covers the four classic points — the centroid, incenter, circumcenter, and orthocenter — what lines build each one, where each one sits, and the mistakes that mix them up

Perpendicular Bisector Theorem: Statement, Proof, and Examples

The perpendicular bisector theorem states that any point on the perpendicular bisector of a segment is equidistant from the segment's two endpoints. This article covers the formal statement, the SAS-based proof, the converse, the link to the circumcentre, and worked examples.

Pairs of Angles: Types, Definitions & Examples

Pairs of angles are two angles linked by a measurement rule or a shared position, the main types being complementary (sum 90°), supplementary (sum 180°), adjacent, vertical, linear pair, and corresponding angles. This guide defines each type, draws the distinctions side by side, and works through six examples.

Opposite Angles: Definition, Theorem & Examples

Opposite angles are the pairs of non-adjacent angles formed when two straight lines intersect, and they are always equal in measure. This guide defines opposite angles, proves the vertically opposite angles theorem, separates them from adjacent and linear-pair angles, and works through six examples

Octagon: Properties, Angles, and Area Formula

An octagon is an eight-sided polygon whose interior angles sum to $1080°$; in a regular octagon each interior angle is $135°$ and each exterior angle is $45°$. This article defines the octagon, derives its area formula $A = 2(1+\sqrt{2}),s^2$, counts its 20 diagonals, and works through examples — starting from the stop sign in every intersection.

Midpoint Formula — Definition, Derivation, and Examples

The midpoint formula finds the exact centre of a line segment by averaging the coordinates of its two endpoints: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. This article defines the midpoint, derives the formula from a simple average, and works through examples with positive, negative, and fractional coordinates.

Lines of Symmetry in a Rectangle — Definition and Examples

A rectangle has exactly 2 lines of symmetry: one vertical line through the midpoints of the longer sides and one horizontal line through the midpoints of the shorter sides. This guide shows why those two folds work, why the diagonals are not lines of symmetry, how a rectangle differs from a square, and how its rotational symmetry adds order 2

Line Symmetry — Definition, Lines of Symmetry, and Examples

Line symmetry means a figure can be folded along a straight line so the two halves match exactly, like a mirror image. This guide defines the line of symmetry, shows how to test for one by folding, gives the line counts for common shapes (square 4, rectangle 2, equilateral triangle 3, circle infinite), and works through examples and mistakes

Intersection of Two Lines: Formula, Methods, and Examples

The intersection of two lines is the single point where they cross, found by solving both line equations together. This guide covers the substitution method, the determinant formula for general-form lines, how to spot parallel lines that never meet, and the mistakes students make most.

Hemisphere: Definition, Volume, Surface Area Formulas, and Examples

A hemisphere is a 3D solid formed when a sphere is cut exactly in half through its centre, leaving one flat circular face and one curved face. Its volume is $\frac{2}{3}\pi r^3$, its curved surface area is $2\pi r^2$, and its total surface area is $3\pi r^2$ — the curved part plus the flat circular base. This article derives all three formulas, explains why the total surface area is not simply half a sphere's, and works through the examples and slips students hit most often

Hyperbola: Definition, Equation, Foci, and Asymptotes

A hyperbola is the set of all points whose distances to two fixed points (the foci) differ by a constant — giving two open curves that mirror each other. This guide covers the standard equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci, vertices, asymptotes, and eccentricity, with the relation $c^2 = a^2 + b^2$, worked examples, and the slips to watch for.

Equilateral Triangle: Definition, Properties, and Formulas

An equilateral triangle is a triangle with all three sides equal and all three angles equal to $60°$. This article covers its definition, properties, the area $\left(\frac{\sqrt{3}}{4}a^2\right)$, perimeter $(3a)$ and height $\left(\frac{\sqrt{3}}{2}a\right)$ formulas with derivations, six worked examples, and common mistakes.

Equiangular Triangles: Definition, Properties, and Examples

An equiangular triangle is a triangle whose three interior angles are all equal, which forces each angle to be 60° and makes it identical to an equilateral triangle. This article covers equiangular triangles in full: the definition, properties, the area and perimeter formulas, why equal angles guarantee equal sides, and the mistakes students make most often.

Difference Between Kite and Rhombus

The main difference between a kite and a rhombus is the sides: a rhombus has all four sides equal, while a kite has two pairs of adjacent equal sides of different lengths. This guide compares them by sides, angles, diagonals, symmetry, and area, and shows why every rhombus is a kite but not every kite is a rhombus.

Coordinate Geometry: Formulas, Concepts & Examples

Coordinate geometry is the branch of math that uses coordinates on a grid to describe points, lines, and shapes, turning geometry into algebra. This guide covers the distance, midpoint, slope, and section formulas, the cartesian plane and its quadrants, and works through six examples that tie them together

Convex Polygon: Definition, Properties, and Examples

A convex polygon is a polygon in which every interior angle is less than $180°$, so no corner caves inward and every diagonal stays inside the figure. This article defines the convex polygon, contrasts it with the concave polygon, gives three reliable tests to tell them apart, and works through examples and common mistakes.

Congruence in Triangles: Meaning, Rules, Examples

Congruence in triangles means two triangles are exactly equal in size and shape — every corresponding side and angle matches, even if one is flipped or turned. This article explains the meaning, the five tests (SSS, SAS, ASA, AAS, RHS), how the $\cong$ symbol and CPCTC work, with six worked examples and common mistakes

Congruent Sides: Definition, Examples, How to Mark

Congruent sides are two or more sides that have exactly the same length. This article explains what congruent sides mean, how tick marks signal them in a diagram, how they classify triangles (isosceles, equilateral) and quadrilaterals, with six worked examples and the mistakes to avoid

Coincident Lines: Definition, Condition & Examples

Coincident lines are two lines that lie exactly on top of each other, sharing every point and therefore infinitely many solutions. This guide defines coincident lines, gives the coefficient-ratio condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, separates them from parallel and intersecting lines, and works six examples

Circumference of Earth — Value, Formula, and How It Was Found

Earth's circumference is about 40,075 km around the equator and 40,008 km around the poles, found from $C = 2\pi r$ using the planet's radius. This guide gives the modern values, the circumference formula, and how Eratosthenes measured it in 240 BCE using only shadow angles and a known distance — landing within about 1% of today's figure

Circumcenter of Triangle - Definition, Formula, Examples

The circumcenter of a triangle is the point where the three perpendicular bisectors of its sides meet. It is equidistant from all three vertices, which makes it the centre of the circumcircle — the one circle that passes through every corner. This article covers the definition, the coordinate formula, the properties, and worked examples

Bisect — Definition, Formula, and Examples

To bisect means to divide something into two equal parts. This article covers what bisect means in geometry, how to bisect a line segment and an angle, the midpoint formula that bisects a segment on a grid, the types of bisectors, and the mistakes students make most often

ASA Congruence Rule: Definition, Proof, Examples

The ASA congruence rule states that two triangles are congruent when two angles and the included side (the side between those two angles) of one match the corresponding parts of the other. This article gives the precise statement, a proof, six worked examples, and the mistakes that trip students up most

Apothem: Definition, Formula, and Examples

The apothem of a regular polygon is the perpendicular distance from its centre to the midpoint of any side. It is found with $a = \dfrac{s}{2\tan(180°/n)}$ and is the key to the area formula $A = \tfrac{1}{2},P,a$. This article defines the apothem, derives both formulas, and works through examples for the hexagon, octagon, and pentagon

270 Degree Angle — Definition, Construction, and Examples

A 270 degree angle is a reflex angle equal to three right angles — three-quarters of a full turn, or $\frac{3\pi}{2}$ radians.

Angles of a Parallelogram: Properties, Theorems, and Examples

In a parallelogram, opposite angles are equal, adjacent (consecutive) angles add to 180°, and all four angles sum to 360°. This guide proves both theorems — opposite angles equal, adjacent angles supplementary — and uses them to find every missing angle, with worked examples, the underlying parallel-line reasoning, and the slips to avoid

3D Geometry Shapes — Types, Properties, and Formulas

3D geometry shapes are solid figures with three dimensions — length, width, and height — so they enclose space and have volume. This guide covers the main types (cube, cuboid, cylinder, cone, sphere, prisms, pyramids), their faces, edges, and vertices, and the surface area and volume formulas for each, with worked examples and the mistakes to avoid

2 Radians to Degrees ≈ 114.59°

2 radians is approximately 114.59° — exactly $\dfrac{360}{\pi}$ degrees. This guide gives the quick answer, a radian-to-degree reference table, where the conversion shows up, the formula behind it, three methods, and the mistakes that trip students up.

Types of Angles — Acute, Right, Obtuse, Reflex

An acute angle measures between $0°$ and $90°$ — but it is only one of five named angle types. This article gives a complete reference table for all five (acute, right, obtuse, straight, reflex), with definitions, diagrams, real-world examples, and the common mistakes students make when sorting them.

60 Degrees to Radians — Value, Formula, and Examples

60 degrees equals $\dfrac{\pi}{3}$ radians, approximately 1.0472. This article shows the conversion formula (multiply degrees by $\frac{\pi}{180}$), a quick reference table of common angles, where $\frac{\pi}{3}$ turns up, the step-by-step method, and the mistakes to avoid.

Projection Vector — Formula, Derivation, and Examples

The projection vector of $\vec{a}$ onto $\vec{b}$ is the "shadow" $\vec{a}$ casts along the direction of $\vec{b}$, given by $\text{proj}_{\vec{b}},\vec{a} = \left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}$. This article covers the projection formula and its derivation, the difference between scalar and vector projection, what a negative projection means, and worked examples.

Collinear Vectors — Definition, Conditions, Examples

Collinear vectors are vectors that lie along the same line — which means one is always a scalar multiple of the other, $\vec{a} = k,\vec{b}$. This article covers the three tests for collinearity (scalar multiple, equal coordinate ratios, zero cross product), how collinear differs from parallel and coplanar, and worked examples.

Vector Addition — Triangle Law, Parallelogram Law, Examples

Vector addition combines two vectors into a single resultant that captures their joint effect. This article covers the three methods — the triangle law, the parallelogram law, and the component method — the resultant-magnitude formula $|\vec{R}| = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}$, and the mistakes students make most.

Vectors in Math — Definition, Types, and Worked Examples

A vector is a quantity with both magnitude and direction, drawn as an arrow whose length is the size and whose arrowhead is the direction. This article covers the types of vectors, how to write them in component form, the core operations (addition, scalar multiples, dot and cross products), and the errors students hit most.

Tetrahedron: Faces, Edges, Vertices, Volume, and Surface Area

A tetrahedron is a 3D solid with 4 triangular faces, 6 edges, and 4 vertices — the simplest possible polyhedron. A regular tetrahedron (all faces equilateral) has volume $\frac{\sqrt{2}}{12}a^3$ and total surface area $\sqrt{3},a^2$, where a is the edge length. This article covers its faces-edges-vertices count, derives the volume and surface-area formulas, shows its net, and works through examples.

Prisms in Geometry: Types, Volume, and Surface Area

A prism is a 3D solid with two identical, parallel polygon bases joined by flat rectangular faces. Its volume is always base area × height and its total surface area is (2 × base area) + (base perimeter × height). This article covers what makes a solid a prism, the main types (triangular, rectangular, pentagonal, and more), and derives the volume and surface-area formulas with worked examples.

Cone: Definition, Volume, Surface Area Formulas, and Examples

A cone is a 3D solid with one circular base that narrows to a single point called the apex. Its volume is $\frac{1}{3}\pi r^2 h$ — exactly one-third of a cylinder with the same base and height — and its total surface area is $\pi r(l + r)$, where l is the slant height. This article derives both formulas, explains the difference between height and slant height, and shows the worked examples and slips that trip students up.

Dodecagon: Definition, Angles, Area, and Properties

A dodecagon is a polygon with 12 sides, 12 vertices, and 12 angles. Its interior angles always add to 1800°, a regular dodecagon has each interior angle equal to 150°, and it carries 54 diagonals. This article covers the definition, the four types, the angle, perimeter, and area formulas with their derivations, and the mistakes students make with the 12-sided shape.

Angles in a Pentagon: Interior, Exterior, and How to Find Them

The interior angles of any pentagon always add to 540°, found with the formula (n − 2) × 180° where n = 5. In a regular pentagon each interior angle is 108° and each exterior angle is 72°. This article shows where 540° comes from, how to find a missing angle in an irregular pentagon, and the slips students make along the way.

Alternate Angles - Definition, Types, Theorem, and Examples

Alternate angles are pairs of angles on opposite sides of a transversal that are equal when the two lines it crosses are parallel. This article defines alternate angles, separates the two types — alternate interior and alternate exterior — states the alternate angles theorem, distinguishes them from co-interior and corresponding angles, and works through six examples.

Degrees - Definition, Symbol, Conversion, and Examples

A degree (°) is the unit we measure angles in — one degree is 1/360 of a complete turn. This article defines degrees, explains the ° symbol and why a full circle holds 360 of them, shows how degrees relate to radians ($180° = \pi$), and works through six examples of measuring, comparing, and converting angles in degrees.

360 Degree Angle - Definition, Shape, and Examples

A 360 degree angle is a full angle (complete angle) — a ray rotates all the way around a vertex and returns to its starting position, sweeping one whole turn. This article defines the 360 degree angle, explains why it equals 2π radians and four right angles, distinguishes it from a 0° angle, and works through six examples involving angles around a point.

180 Degree Angle - Definition, Shape, and Examples

A 180 degree angle is a straight angle — its two arms point in exactly opposite directions from a shared vertex, forming a perfectly straight line. This article defines the 180 degree angle, explains why it equals a half turn and π radians, distinguishes it from a straight line, and works through six examples involving supplementary angles and triangles.

Right Angle - Definition, Properties, and Worked Examples

A right angle is an angle that measures exactly 90° — a quarter turn, marked with a small square at the vertex instead of an arc. This article defines the right angle, shows how to identify and verify one, explains why it sits at the centre of perpendicular lines and the right-angled triangle, and works through six examples.

Equation of a Straight Line — Forms & Examples

The equation of a straight line is a relationship between $x$ and $y$ that every point on the line satisfies, written most commonly as y = mx + c (slope-intercept) or Ax + By + C = 0 (standard form). This article covers all five major forms — slope-intercept, point-slope, standard, intercept, and two-point — when to use each, how to convert between them, and six worked examples.

Point Slope Form — Formula, Derivation, Examples

Point slope form writes the equation of a line as y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any known point on the line. This article covers the formula, its derivation straight from the slope definition, how to convert it to slope-intercept and standard form, six worked examples, and the sign mistakes students make most.

Finding Slope From Two Points — Formula & Examples

To find the slope from two points $(x_1, y_1)$ and $(x_2, y_2)$, use the formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ — the change in y over the change in x, read as "rise over run." This article covers the formula, where it comes from, a clean step-by-step method, six worked examples, and the order-of-subtraction mistake that trips up most students.

Intersecting Lines — Definition, Angles, Examples

Intersecting lines are two or more lines that cross at exactly one common point, called the point of intersection, and they form four angles there — two pairs of equal vertical angles and pairs of supplementary adjacent angles. This article covers the definition, the properties, the angles formed, perpendicular and oblique cases, six worked examples, and the mistakes students make.

Vertical Line — Equation, Slope, and Examples

A vertical line is a straight line where every point shares the same x-coordinate, so its equation is x = a and its slope is undefined (the run is zero, and you cannot divide by zero). This article covers the definition, the equation, why the slope is undefined, the vertical line test, six worked examples, and the errors students make most.

Foci of Hyperbola — Formula, How to Find, and Examples

The foci of a hyperbola are two fixed points whose difference of distances to any point on the curve is constant. They are found from $c^2 = a^2 + b^2$, sitting at $(\pm c, 0)$ for a horizontal hyperbola centered at the origin. This article covers the focus formula, the defining difference-of-distances property, eccentricity, and worked examples on the foci of a hyperbola — note this differs from the ellipse, where $c^2 = a^2 - b^2$.

Area of Ellipse — Formula, Derivation, and Examples

The area of an ellipse is $A = \pi a b$, where $a$ is the semi-major axis and $b$ is the semi-minor axis (the half-lengths of the longest and shortest diameters). This article covers the area of ellipse formula, two ways to derive it — by stretching a circle and by integration — a variable glossary, and worked examples. When $a = b$, the ellipse becomes a circle and the formula collapses to $\pi r^2$.

Semicircle — Definition, Area, and Perimeter Formula

A semicircle is exactly half a circle, formed by cutting along a diameter. Its area is $\dfrac{\pi r^2}{2}$, but its perimeter is $\pi r + 2r$ — not half the circle's perimeter, because the straight diameter edge counts too. This article covers both formulas, the angle in a semicircle, and worked examples.

Secant of a Circle — Definition, Formula, and Examples

A secant of a circle is a straight line that intersects the circle at two distinct points — it is a chord extended past both ends. This article covers the secant definition, how it differs from a chord and a tangent, the two power-of-a-point theorems ($PA \cdot PB = PC \cdot PD$ and the tangent–secant relation), and worked examples of a secant of a circle.

Circles in Geometry — Parts, Formulas, and Examples

A circle is the set of all points in a plane that sit the same distance from a fixed center, and that fixed distance is the radius. This hub walks through every part of a circle — radius, diameter, chord, secant, tangent, arc, sector, segment — and the two formulas that do most of the work: circumference $C = 2\pi r$ and area $A = \pi r^2$.

What is a Diagonal - Definition, Formula, and Examples

A diagonal is a straight line segment that joins two non-adjacent vertices (corners) of a polygon, never a side. The number of diagonals in any polygon with $n$ sides is given by the formula $\dfrac{n(n-3)}{2}$. This guide defines the diagonal, derives the counting formula, and works through examples for triangles up to hexagons and 3D solids.

Difference Between a Square and a Rectangle - Properties Compared

The core difference between a square and a rectangle is the sides: a square has all four sides equal, while a rectangle has only its opposite sides equal. Both have four right angles and equal diagonals, but a square's diagonals cross at 90° and a rectangle's do not. This guide compares every property with formulas and examples.

Diagonal of Rhombus — Formula, Properties, Examples

The two diagonals of a rhombus bisect each other at right angles, splitting it into four congruent right triangles. That single fact gives you every diagonal formula: from area, $p = \dfrac{2 \times \text{Area}}{q}$, and from the side, $\left(\dfrac{p}{2}\right)^2 + \left(\dfrac{q}{2}\right)^2 = a^2$. This guide derives both and works through six examples.

Properties of Parallelogram — Sides, Angles, Diagonals

A parallelogram is a quadrilateral with both pairs of opposite sides parallel, and that single rule forces every other property: opposite sides are equal, opposite angles are equal, consecutive angles add to 180°, and the diagonals bisect each other. This guide proves each property, lists the formulas, and works through six examples.

Quadrilaterals — Types, Properties, and Formulas

A quadrilateral is a closed, four-sided polygon whose four interior angles always add up to 360°. This guide maps the whole family (parallelogram, rectangle, square, rhombus, trapezium, and kite), showing how each one inherits or breaks the rules of the others, with their diagonals, area formulas, and worked examples.

Equidistant in Geometry — Meaning, Formula, Examples

Equidistant means being at an equal distance from two or more points, lines, or objects. This article gives the plain meaning, the distance and midpoint formulas that test for it, the role of the perpendicular bisector (every point on it is equidistant from a segment's endpoints), and worked examples — plus where equidistance hides inside circles and triangle centers.

Incenter of a Triangle — Properties, Formula, Examples

The incenter of a triangle is the single point where its three angle bisectors meet, and it sits equidistant from all three sides — making it the center of the largest circle that fits inside the triangle. This article covers the definition, the key properties, the coordinate and angle formulas, the inradius, and worked examples, plus how the incenter differs from the circumcenter.

Geometrical Proofs — Two-Column, Paragraph & Flow Proofs

A geometrical proof is a step-by-step argument where every claim about a figure is backed by a reason — a given fact, definition, postulate, or theorem — until the goal is reached. This article teaches the three formats — two-column, paragraph, and flowchart — shows the statement-and-reason structure, and works through full proofs you can copy as templates.

Side Angle Side (SAS) — Congruence and Similarity Rules

Side Angle Side (SAS) is one criterion that does two jobs: two sides and the included angle prove triangles congruent when the sides are equal, and similar when the sides are proportional. This article keeps the two apart — same angle condition, different side condition — with a labelled diagram and proof for each, plus worked examples and the mistakes that blur them.

Triangle Congruence Theorem — SSS, SAS, ASA, AAS, RHS

A triangle congruence theorem is a shortcut rule that confirms two triangles are identical in shape and size by checking only three matching parts instead of all six. This article walks through the five rules — SSS, SAS, ASA, AAS, and RHS — with a labelled diagram for each, worked examples, the AAA and SSA cases that fail, and how to pick the right rule fast.

Isosceles Triangle Theorem: Proof, Converse, Examples

The isosceles triangle theorem says that if two sides of a triangle are equal, then the angles opposite those sides — the base angles — are also equal. Its converse runs the other way: equal base angles force the opposite sides to be equal. This article gives the statement, a full proof using triangle congruence, the converse and its proof, and six worked examples.

Height of Equilateral Triangle: Formula, Proof, Examples

The height of an equilateral triangle with side $a$ is $h = \dfrac{\sqrt{3}}{2},a$ — the altitude from any vertex straight down to the opposite side. It comes from the Pythagoras theorem applied to the right triangle the altitude creates. This article gives the formula, its derivation, methods for finding height from the side, area, or perimeter, and six worked examples.

Exterior Angles of Triangle: Theorem, Formula, Examples

An exterior angle of a triangle is formed by extending one side, and it equals the sum of the two non-adjacent (remote) interior angles. Each exterior angle is also supplementary to its adjacent interior angle, and the three exterior angles taken one per vertex sum to 360°. This article covers the exterior angle theorem, the formulas, a short proof, and six worked examples.

Properties of a Triangle: Formulas, Theorems, Examples

A triangle has three sides, three vertices, and three angles, and its core properties are: the interior angles sum to 180°, any two sides together exceed the third, and the largest angle faces the longest side. This article walks through every key property of a triangle — the angle sum property, triangle inequality, exterior angle property, side-angle relationship, Pythagoras property, and congruence — with formulas and six worked examples.

Obtuse Triangle - Definition, Properties, Formulas

An obtuse triangle is a triangle with exactly one angle greater than 90°, and that angle sits opposite the longest side. A triangle is obtuse when the square of its longest side beats the sum of the squares of the other two: $c^2 > a^2 + b^2$. This article gives the definition, the properties, the area and Heron's formula, the side test, and six worked examples.

Angle Bisector Theorem: Statement, Proof, Examples

The angle bisector theorem states that the bisector of an angle in a triangle divides the opposite side into two segments whose ratio equals the ratio of the two adjacent sides: BD/DC = AB/AC. This article covers the statement, the internal and external versions, a clean similar-triangles proof, the converse, the length-of-bisector formula, six worked examples, the common mistakes, and where the theorem leads next.

Difference Between a Line and a Line Segment

The difference between a line and a line segment comes down to ends: a line has no endpoints and runs infinitely in both directions, while a line segment has two endpoints and a definite, measurable length. This article covers both definitions, a side-by-side comparison table, the notation, real-world examples, and the common mistakes students make telling them apart.

Cross Product of Two Vectors: Formula, Rule, Examples

The cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ produces a third vector perpendicular to both, written $\mathbf{a} \times \mathbf{b}$, with magnitude $|\mathbf{a}||\mathbf{b}|\sin\theta$ and direction set by the right-hand rule. This article covers the formula, the determinant method of computing it, the right-hand rule, the area-of-a-parallelogram link, and how it differs from the dot product.

Eccentricity of an Ellipse: Formula, Meaning, Examples

The eccentricity of an ellipse is a single number, $e = \dfrac{c}{a}$, that measures how stretched the ellipse is, from a perfect circle to a long, thin oval. This article covers what eccentricity measures, the two equivalent formulas $e = \dfrac{c}{a}$ and $e = \sqrt{1 - \dfrac{b^2}{a^2}}$, why it always lies between 0 and 1, the derivation, and worked examples.

Center of a Circle: Definition, Formula, How to Find

The center of a circle is the single fixed point that sits the same distance, the radius, from every point on the circle. This article covers the definition, the equation form $(x - h)^2 + (y - k)^2 = r^2$, and three ways to find the center: from the equation, from the endpoints of a diameter, and from three points on the circle.

Parallel Lines Cut by a Transversal: Angles, Properties, Examples

When two parallel lines are cut by a transversal, the crossing makes eight angles that fall into four named pairs — corresponding, alternate interior, alternate exterior, and co-interior angles — and the parallel condition forces each pair to be either equal or supplementary. This article maps the full configuration, defines all eight angles and every pair, lays out the properties, works six examples solving for x, and flags the common mistakes.

Obtuse Scalene Triangle: Properties & Examples

An obtuse scalene triangle is a triangle with one obtuse angle and all three sides of different lengths — combining the "obtuse" classification by angle with the "scalene" classification by side. This article covers the definition, the properties, how to find its area by base-and-height and by Heron's formula, real-world examples, six worked problems, and the common mistakes.

Angle Side Angle (ASA): Rule, Proof, Examples

The angle side angle (ASA) rule states that two triangles are congruent if two angles and the side included between them in one triangle equal the corresponding two angles and included side of the other — and the word included is what separates ASA from AAS. This article covers the statement, a full proof, two-column proof use, the ASA-versus-AAS difference, six worked examples, and the common mistakes.

Segment Addition Postulate: Formula & Examples

The segment addition postulate states that if point B lies between points A and C on a line segment, then AB + BC = AC — the two shorter parts add to the whole. This article covers the definition, the formula, why "between" matters, how it finds a midpoint, how it powers two-column proofs, six worked examples, and the mistakes to avoid.

Transitive Property of Congruence: Examples

The transitive property of congruence states that if one figure is congruent to a second and the second is congruent to a third, then the first is congruent to the third — in symbols, if a ≅ b and b ≅ c, then a ≅ c. This article covers the statement for segments, angles, and triangles, where it differs from the substitution property, two-column proofs that use it, six worked examples, and the mistakes to watch for.

What Is a Polyhedron? Types & Euler's Formula

A polyhedron is a three-dimensional solid bounded entirely by flat polygon faces, joined along straight edges that meet at points called vertices. This article defines the polyhedron, names its parts, walks through the types, and shows how Euler's formula, $F + V - E = 2$, ties faces, vertices, and edges together.

Irregular Polygons: Definition, Types & Area

Irregular polygons are closed flat shapes whose sides are not all equal and whose angles are not all equal — the opposite of regular polygons. This article defines them, lists the common types, shows that the interior-angle sum is still $(n-2)\times 180°$, and walks through finding the area by decomposition — splitting the shape into triangles and rectangles.

Exterior Angles of a Polygon: Sum & Formula

The exterior angles of a polygon are the angles between each side and the extension of its neighbour, and they always sum to $360°$ — for any polygon, regular or irregular, no matter how many sides. In a regular polygon, each exterior angle is $\dfrac{360°}{n}$. This article defines the exterior angle, derives the $360°$ sum, gives the formula, and works through examples.

Coterminal Angles: Definition, Formula, and Examples

Coterminal angles are angles in standard position that share the same terminal side — they land in the same place after differing by a whole number of full turns. You find them by adding or subtracting $360°$ (or $2\pi$ radians). This article covers the definition, the formula in degrees and radians, positive and negative coterminal angles, the unit-circle picture, and worked examples.

Obtuse Angle: Definition, Degrees & Examples

An obtuse angle is any angle that measures more than $90°$ and less than $180°$ — wider than a right angle but not yet a straight line. This article defines the obtuse angle, lists its properties, shows where it appears in real life and in triangles, and clears up the mistakes that trip students up.

X Intercept: Definition, Formula & Examples

Ordered Pair: Definition, (x, y) Notation & Examples

An ordered pair is two numbers written $(x, y)$ in a fixed order that together name a single point, where the order matters: $(2, 4)$ and $(4, 2)$ are different points. This article covers the notation, how to plot an ordered pair, the equality rule, where order changes the answer, and the mistakes students make most.

Coordinate Plane: Plot Points, Quadrants & Examples

A coordinate plane is the grid where every point is located by an ordered pair $(x, y)$ — the x-coordinate first, the y-coordinate second. This article shows how to plot a point, how to read a point's coordinates off a graph, the four quadrants and their sign rules, and the mistakes students make most.

Cartesian Plane: Definition, History & Quadrants

The Cartesian plane is the two-axis coordinate system invented by René Descartes that lets every point be named by an ordered pair $(x, y)$. This article covers who built it and why, how the plane is constructed from two perpendicular axes and an origin, the four quadrants, and worked examples of locating points.

Slope of Parallel Lines: Formula & Examples

The slope of parallel lines is the same for both lines: if two lines are parallel, then $m_1 = m_2$. This article explains why equal slopes force two lines to stay parallel, derives the rule, works through six examples, and clears up the mistakes that trip students up most.

Side Side Side (SSS): Congruence Proof & Examples

The side side side (SSS) rule states that if the three sides of one triangle equal the three sides of another, the triangles are congruent — with no angle information required at all. This article covers the statement, why three sides lock a triangle's shape, the proof, SSS similarity, six worked examples, the common mistakes, and where the rule leads next.

Points and Lines: Definition & Examples

Points and lines are the two most basic ideas in geometry: a point marks an exact position with no size, and a line is an endless, straight row of points. This article covers how a line is built from points, collinear versus non-collinear points, the incidence rules that connect them, and six worked examples.

Square: Properties, Area, Perimeter, Diagonal

A square is a quadrilateral with four equal sides and four right angles — the most regular four-sided shape there is. This article covers its properties and the three formulas, all derived from the shape itself: area $A = s^2$, perimeter $P = 4s$, and diagonal $d = s\sqrt{2}$, with six worked examples and the mistakes students make most.

Straight Line: Definition, Properties & Slope

A straight line is a one-dimensional figure that extends infinitely in both directions, has no curves, and keeps a constant slope throughout. This article covers the definition and properties, what slope means and how to find it, a brief tour of the equation forms, and six worked examples.

Scale in Maths: Scale Drawings and Map Scale

In maths, scale is the ratio between a length on a drawing or map and the matching length in real life, written like 1 cm : 5 km or 1 : 50,000. This article covers what scale means, how to read a map scale, how to convert between map distance and real distance in both directions, six worked examples, and the mistakes students make most.

Unit Circle: Definition, Coordinates, Chart, Examples

The unit circle is the circle of radius 1 centred at the origin, where every point on the rim has coordinates (cos θ, sin θ) for the angle θ measured from the positive x-axis. This article covers the definition, the equation x² + y² = 1, why the coordinates are cosine and sine, the special angles in both degrees and radians, six worked examples, and the common mistakes.

Rectangular Pyramid: Volume, Surface Area, Faces

A rectangular pyramid is a 3D solid with a rectangular base and four triangular faces that meet at a single apex — giving 5 faces, 8 edges, and 5 vertices. This article covers its properties, the volume formula $V = \tfrac{1}{3} \times l \times w \times h$, the surface area formulas, the net, six worked examples, and the mistakes students make most.

AAS Congruence Rule: Proofs, AAS vs ASA

The AAS congruence rule lets you prove two triangles congruent when two angles and a non-included side of one match the other — and the trick to using it is reading the figure to confirm the side sits outside the two angles. This article is a how-to-apply guide: when AAS is the right call, how to write the two-column proof, AAS versus ASA, six worked proof exercises, and the mistakes to avoid.

Hypotenuse Leg Theorem (HL): Proof & Examples

The hypotenuse leg theorem (HL) states that two right triangles are congruent if their hypotenuses are equal and one pair of legs is equal — just two matching sides, not three. This article covers the statement, why it works only for right triangles, the RHS link, the Pythagorean proof, six worked examples, the common mistakes, and where the rule leads next.

Reference Angle: Definition, Formulas, Examples

A reference angle is the positive acute angle between the terminal side of an angle and the x-axis, always between $0^\circ$ and $90^\circ$ ($0$ and $\tfrac{\pi}{2}$). This article covers the definition, the per-quadrant rules in both degrees and radians, how to handle negative and large angles, why reference angles let you evaluate trig functions anywhere, six worked examples, and the common mistakes.

Alternate Interior Angles Theorem: Proof

The alternate interior angles theorem states that when a transversal crosses two parallel lines, each pair of alternate interior angles is congruent (equal). This article gives the formal statement, a full two-step proof, the converse and its proof, the related co-interior angles theorem, and six worked examples. For the underlying definition of the angle pair, see Alternate Interior Angles.

Acute Scalene Triangle: Properties & Examples

An acute scalene triangle has all three angles less than 90° and all three sides of different lengths, so no two angles and no two sides ever match. This article covers the definition, how a triangle can be acute and scalene at once, the properties, the area and perimeter formulas with derivation, six worked examples, and the common mistakes.

Difference Between Rhombus and Rectangle

The difference between a rhombus and a rectangle is that a rhombus has four equal sides but slanted angles, while a rectangle has four right angles but only its opposite sides equal. This article covers both shapes' properties in full, what they share as parallelograms, the key differences in a comparison table, when each one matters, and six worked examples.

Radians to Degrees — Conversion Table and Formula

To convert radians to degrees, multiply by $180°/\pi$. The formula: $\text{degrees} = \text{radians} \times \dfrac{180°}{\pi}$. This article gives a complete conversion table for every common angle (multiples of $\pi/12$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$ and beyond), three worked examples, the reverse direction, and the most common mistakes.

Scale Factor: Definition, Formula & Examples

The scale factor is the number you multiply every length of a figure by to get the matching length of a similar figure, equal to new length ÷ original length. This article covers the formula, scaling up versus down, dilation on the coordinate plane, the area ($k^2$) and volume ($k^3$) rules, and six worked examples.

Dilation in Geometry: Definition, Scale Factor, Examples

A dilation in geometry is a transformation that resizes a figure about a fixed center of dilation by a scale factor, making it larger or smaller while keeping its shape exactly the same. This article covers the definition, the coordinate rule (x, y) → (kx, ky), how to find and use the scale factor, what happens with fractions and negative values, six worked examples, and the common mistakes.

Symmetry in Geometry - Types, Definition, Examples

Symmetry in geometry means a shape looks identical after being transformed — moved, rotated, or flipped. There are three core types: reflection symmetry (mirror image across a line), rotational symmetry (looks the same after rotation by a fixed angle), and point symmetry (every point has a matching point through a central point)

Geometric Transformations: Definition, Types and Examples

Foci of an Ellipse: Definition, Formula, Examples

The foci of an ellipse are two fixed points on its major axis such that, for any point on the ellipse, the sum of its distances to the two foci is constant (equal to 2a). This article covers the definition, where the foci sit, the formula c² = a² − b², six worked examples, and the common mistakes.

Ellipse - Equation, Formula, Properties, Graphing

An ellipse is the set of all points in a plane whose distances to two fixed points (called foci) sum to a constant. Its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Directrix of a Parabola: Definition, Equation, Examples

The directrix of a parabola is a fixed line, perpendicular to the axis, such that every point on the curve is exactly as far from the directrix as it is from the focus. This article covers the equidistance definition, the directrix equation for all four standard forms ($y^2 = 4ax$ gives directrix $x = -a$), how to find and derive it, six worked examples, and the common mistakes.

Focus of a Parabola: Definition, Formula, Examples

The focus of a parabola is the single fixed point on the axis such that every point on the curve is the same distance from the focus as it is from the directrix line. This article covers the definition, the focus formula for all four standard forms ($y^2 = 4ax$ gives focus $(a, 0)$), the focal distance, the latus rectum and focal chord, six worked examples, and the common mistakes.

Parabola - Definition, Formula, Graph, Examples

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). It's one of four classical conic sections — created by slicing a cone with a plane parallel to its slant side. The standard equation is $y^2 = 4ax$ (horizontal opening) or $(x - h)^2 = 4p(y - k)$ (vertex form).

Conic Sections: Types, Formulas & Equations

A conic section is the curve you get when a flat plane slices through a cone, and tilting the slice produces exactly four shapes: the circle, ellipse, parabola, and hyperbola. This article covers the definition, the four types, their eccentricity values, the focus-directrix idea, standard equations, six worked examples, and the mistakes students make most.

Parallel & Perpendicular Lines: Slope Rules

Parallel and perpendicular lines are two relationships you can read straight off slopes: parallel lines have equal slopes ($m_1 = m_2$), and perpendicular lines have slopes that multiply to −1 ($m_1 \cdot m_2 = -1$). This article covers both slope rules, why each works, how to tell lines apart from their equations, the special vertical-line case, and six worked examples.

Slope of Perpendicular Lines: The −1 Rule

The slope of perpendicular lines follows one rule: the product of the two slopes is −1, so $m_1 \cdot m_2 = -1$. This article covers the negative-reciprocal rule, why it holds, how to find a perpendicular slope from any given slope, the vertical-horizontal exception, and six worked examples.

Undefined Slope: Definition, Equation & Examples

An undefined slope is the slope of a vertical line, where every point shares the same x-coordinate, so the slope formula divides by zero. This article covers why a vertical line's slope is undefined, the equation x = a, the graph, how undefined slope differs from zero slope, and six worked examples.

Horizontal Line - Definition, Equation, and Slope

A horizontal line is a straight line that runs parallel to the x-axis. Its equation has the form $y = b$ (where $b$ is a constant), its slope is exactly $\mathbf{0}$, and it intersects the y-axis at the single point $(0, b)$.

Intercept Form of a Line: Formula & Examples

The intercept form of a line is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept (where the line crosses the x-axis) and $b$ is the y-intercept (where it crosses the y-axis). This article covers the formula, its derivation, how to read or graph a line straight from its intercepts, the triangle it cuts with the axes, and six worked examples.

Y Intercept: Definition, Formula & Examples

The y intercept is the point where a graph crosses the y-axis, found by setting $x = 0$ and solving for $y$. For a line written as $y = mx + b$, the y intercept is simply $b$. This article covers the definition, the method for every equation form, the parabola case, six worked examples, and the mistakes students make most.

y = mx + b: Read Slope & Y-Intercept, Plot a Line

In the equation $y = mx + b$, the number $m$ is the slope (how steeply the line rises or falls) and the number $b$ is the y-intercept (where the line crosses the y-axis). This article shows how to read $m$ and $b$ straight off the equation, plot the line, write the equation from points or a graph, and rearrange any linear equation into this form.

Slope Intercept Form: y = mx + b & Examples

Slope intercept form is the equation of a straight line written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This article covers how to read $m$ and $b$ straight off the equation, how to find them from points or a graph, how to convert other forms, and six worked examples.

Slope of a Line - Formula, Calculation, Examples

The slope of a line - sometimes called the gradient — measures the line's steepness as the ratio of vertical change to horizontal change between any two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$, or "rise over run."

Quadrants of Coordinate Plane - I, II, III, IV

The coordinate plane is divided by the x-axis and y-axis into four quadrants, numbered I, II, III, IV counterclockwise starting from the upper right. Each quadrant has a specific sign convention for $(x, y)$: Quadrant I: both positive; II: $x$ negative, $y$ positive; III: both negative; IV: $x$ positive, $y$ negative.

X and Y Axis in a Graph: Origin & Quadrants

In a graph, the x-axis is the horizontal number line and the y-axis is the vertical number line; they cross at the origin, the point (0, 0). This article explains the two axes, the four quadrants and their sign conventions, and how to plot any point from its ordered pair $(x, y)$, with six worked examples.

Triangular Pyramid: Volume, Surface Area & Faces

A triangular pyramid, also called a tetrahedron, is a 3D solid with 4 triangular faces, 6 edges, and 4 vertices — the simplest possible polyhedron. This article covers its properties, the volume formula $V = \tfrac{1}{3} \times \text{base area} \times \text{height}$, the surface area formulas, the net, six worked examples, and the mistakes students make most.

Triangular Prism - Volume, Surface Area, Formulas

A triangular prism is a 3D solid with 2 triangular bases and 3 rectangular lateral faces — total 5 faces, 9 edges, 6 vertices. The volume is $V = (\text{area of triangle}) \times L = \tfrac{1}{2}bh \times L$ where $b, h$ are the triangle's base and height, and $L$ is the prism's length. The surface area = sum of the two triangle areas + the three rectangle areas.

Rectangular Prism - Volume, Surface Area, Formulas

A rectangular prism (also called a cuboid) is a 3D solid with 6 rectangular faces, 12 edges, and 8 vertices. Its volume is $V = l \times w \times h$ (length × width × height), and its surface area is $S = 2(lw + lh + wh)$.

Cylinder — Shape, Formula, Examples

Central Angle in Geometry: Definition, Formula, Examples

A central angle in geometry is an angle whose vertex sits at the centre of a circle and whose two sides are radii, and its measure equals the measure of the arc it cuts off. This article covers the definition, the formula in both degrees and radians, the central angle theorem (the central angle is twice an inscribed angle on the same arc), six worked examples, and the common mistakes.

Tangent in Geometry — Definition, Formula, Examples

A tangent is a straight line that touches a curve at exactly one point and does not cross it there. This article covers the geometric tangent (focused on the circle), the formulas that describe a tangent line, the two foundational tangent theorems, three worked examples (Quick, Standard, Stretch), and the mistakes students make most often.

Equation of a Circle: Standard & General Form

The equation of a circle in standard form is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This article covers the standard and general forms, the derivation straight from the Pythagorean theorem, how to read the centre and radius off either form, how to convert between them by completing the square, and six worked examples.

Arc Length: Formula, How to Find It, Examples

Arc length is the distance measured along the curved edge of a circle, a fraction of the full circumference set by the central angle: L = (θ/360°) × 2πr in degrees, or L = rθ in radians. This article covers the definition, both formulas, the derivation from the circumference, how to find arc length with and without the angle, six worked examples, and the common mistakes.

Sector of a Circle: Area, Arc Length, Perimeter

A sector of a circle is the pie-slice region enclosed by two radii and the arc between them, and its area is the fraction of the whole circle set by its central angle: Area = (θ/360°) × πr². This article covers the definition, minor and major sectors, the area, arc length, and perimeter formulas in both degrees and radians, six worked examples, and the common mistakes.

Area of a Circle: Formula, Derivation & Examples

The area of a circle is the flat space enclosed inside its boundary, given by the formula $A = \pi r^2$, where $r$ is the radius. This article defines the area, derives πr² by unrolling the circle into a triangle, covers area from the diameter and circumference, and works through six examples.

Circumference of a Circle - Formula, Examples

The circumference of a circle is the distance around it — its perimeter. Given the radius $r$, the formula is $C = 2\pi r$. Given the diameter $d = 2r$, equivalently $C = \pi d$. The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter — a universal property of all circles.

Chord of a Circle: Formula, Theorems, Examples

A chord of a circle is a straight line segment joining any two points on the circle's boundary, and the longest possible chord is the diameter. This article covers the definition, the two formulas for chord length (from the perpendicular distance and from the central angle), the main chord theorems with proof, six worked examples, and the common mistakes.

Diameter of a Circle — Formula, Worked Examples, and Properties

The diameter of a circle is the longest chord, passing through the centre, equal to twice the radius. Formula: $d = 2r$, or $d = C/\pi$ from circumference, or $d = 2\sqrt{A/\pi}$ from area. This article gives the three diameter formulas, three worked examples (Quick, Standard, Stretch), and the historical thread from Archimedes to modern usage.

Parts of a Circle: Names, Definitions & Diagram

The main parts of a circle are the centre, radius, diameter, circumference, chord, arc, sector, segment, tangent, and secant, every one of them defined by its relationship to the single fixed centre point. This article names and explains each part with a labelled diagram, the formulas tied to them, six worked examples, and the common mistakes students make.

Interior Angles: Sum Formula & Examples

Interior angles are the angles inside a polygon, one at each vertex. The sum of all of them is $(n-2) \times 180°$ for an $n$-sided polygon, and in a regular polygon each one equals that sum divided by $n$. This article covers the definition, the sum formula and where it comes from, regular versus irregular polygons, the interior angles between parallel lines, and six worked examples.

Hexagon Shape — Definition, Types, Properties, and Area Formula

Hexagon is a six-sided closed two-dimensional polygon with six vertices and six interior angles. In a regular hexagon, all six sides are equal, all six interior angles are $120°$, and the sum of interior angles is $720°$. The area of a regular hexagon with side $s$ is $\frac{3\sqrt{3}}{2}s^2$.

Pentagon Shape - Properties, Area, and Perimeter

A pentagon is a polygon with 5 sides and 5 interior angles summing to $540°$. A regular pentagon has all sides equal and all angles equal to $108°$ each. Its area formula is $A = \tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2 \approx 1.72 s^2$, and its perimeter is $P = 5s$.

Shapes in Geometry — Complete 2D and 3D Taxonomy

Geometric shapes split into 2D (flat, with length and width) and 3D (solid, with length, width, and height). 2D shapes group into polygons (triangles, quadrilaterals, regular and irregular) and non-polygons (circle, ellipse). 3D shapes group into polyhedra (prisms, pyramids, Platonic solids) and curved solids (sphere, cylinder, cone).

Geometric Shapes: Types, Properties & Examples

Geometric shapes are closed figures built from points, lines, and curves, and they split into two families: flat 2D shapes and solid 3D shapes. This article covers the full list of types, the properties that separate one shape from the next, the area and volume formulas, six worked examples, and where shapes show up around you.

Difference Between Square and Rhombus Explained

The difference between a square and a rhombus is that a square has four right angles and equal-length diagonals, while a rhombus has four equal sides but its angles are not 90° and its diagonals are unequal. This article covers both shapes' properties in full, what they share, the key differences in a comparison table, when each one matters, and six worked examples.

Is a Square a Rectangle? Yes — Here's Why

Yes — every square is a rectangle, because a rectangle is defined as a quadrilateral with four right angles, and a square has those four right angles plus the extra condition that all its sides are equal. This article explains the definitions, the quadrilateral family tree, why a square is a special rectangle, why the reverse is not always true, six examples, and the common mistakes.

Properties of a Kite: Sides, Angles & Diagonals

A kite is a quadrilateral with two pairs of adjacent equal sides, diagonals that cross at right angles, and one pair of equal opposite angles. This article covers every property of a kite by sides, angles, diagonals, and symmetry, derives the area formula $\tfrac{1}{2} \times d_1 \times d_2$, and works through six examples.

Properties of a Rectangle: Sides, Angles & Diagonals

A rectangle is a quadrilateral with four right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other. This article covers every property of a rectangle by sides, angles, and diagonals, derives the area $l \times w$, perimeter $2(l+w)$, and diagonal $\sqrt{l^2 + w^2}$ formulas, and works through six examples.

Isosceles Trapezoid: Properties, Area & Examples

An isosceles trapezoid is a four-sided shape with one pair of parallel sides (the bases) and two non-parallel sides (the legs) of equal length, which gives it equal base angles, equal diagonals, and a line of symmetry. This article covers its definition, properties, the area formula $A = \tfrac{1}{2}(a+b)h$, perimeter, diagonals, six worked examples, and where students go wrong.

Trapezium - Definition, Properties, Area and Examples

A trapezium (US: trapezoid) is a quadrilateral with one pair of parallel sides. The parallel sides are called bases; the non-parallel sides are legs. The area formula is $A = \tfrac{1}{2}(a + b) \cdot h$ — the average of the parallel sides times the height.

Trapezoid: Properties, Area, and Formula Guide

Rhombus: Properties, Area, and Perimeter

Parallelogram - Properties, Area, and Formulas

Angle Angle Side (AAS) Congruence: Proof, Examples

Angle Angle Side (AAS) is a triangle congruence rule: if two angles and a non-included side of one triangle equal the corresponding two angles and side of another, the triangles are congruent. This article covers the statement, why it works, the proof from ASA, the difference between AAS and ASA, six worked examples, and the common mistakes.

CPCTC: Meaning, Proof & Examples

CPCTC stands for corresponding parts of congruent triangles are congruent: once two triangles are proven congruent, every matching pair of their sides and angles is automatically equal. This article covers what CPCTC means, why it follows from the definition of congruence, how it works as the final step of a two-column proof, six worked examples, the common mistakes, and where it leads next.

Congruent (Congruence) - Meaning, Definition, Examples

Congruent means identical in shape AND size. Two figures are congruent if one can be transformed into the other by rigid motions — translation, rotation, reflection — without stretching or shrinking. The symbol is $\cong$. For triangles, the five congruence theorems (SSS, SAS, ASA, AAS, RHS) let you prove congruence without measuring every side and angle.

Similar Triangles: Theorems & Properties

Similar triangles are triangles with the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio. This article covers the definition, the AA, SAS, and SSS similarity criteria, the properties, the area-ratio rule, the difference from congruent triangles, six worked examples, and the mistakes students make most.

Median of a Triangle: Properties & Formula

The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, so it always cuts that side into two equal halves. This article covers its definition, properties, the centroid and its 2:1 ratio, the length formula from Apollonius's theorem, six worked examples, and the mistakes students make most.

Altitude of a Triangle: Formulas & Properties

The altitude of a triangle is the perpendicular segment from a vertex straight down to the line containing the opposite side, and its length is the height used in the area formula. This article covers the definition, the formulas for scalene, isosceles, equilateral, and right triangles, the orthocentre, six worked examples, and the mistakes students make most.

Exterior Angle Theorem: Statement, Proof, Examples

The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two interior angles not next to it: exterior angle = sum of the two remote interior angles. This article covers the statement, a clean proof, the exterior angle inequality version, six worked examples, the common mistakes, and where the theorem leads.

Triangle Sum Theorem: Proof, Formula, Examples

The triangle sum theorem states that the three interior angles of any triangle always add up to 180° — written ∠A + ∠B + ∠C = 180°. This article covers the statement, the parallel-line proof, the exterior-angle link, why the rule holds only in flat (Euclidean) space, six worked examples, the common mistakes, and where the theorem leads next.

Pythagoras Theorem - Formula, Proof, Examples

Pythagoras theorem says a² + b² = c². Learn the formula, four proofs, common mistakes, the 4,000-year-old Babylonian tablet, and worked examples.

Right Scalene Triangle: Properties & Examples

A right scalene triangle has one right angle (90°) and three sides all of different lengths, which also forces its three angles to be different. This article covers the definition, how a triangle can be right and scalene at once, the properties, the area and perimeter formulas with derivation, six worked examples, and the common mistakes.

Right Angled Triangle: Properties & Formulas

A right angled triangle is a triangle with one angle of exactly 90°, and the side opposite that angle, the hypotenuse, is always the longest. This article covers its definition, properties, the Pythagorean theorem, area and perimeter formulas, the special 45-45-90 and 30-60-90 types, six worked examples, and the mistakes students make most.

Isosceles Obtuse Triangle: Properties & Examples

An isosceles obtuse triangle has one obtuse angle (between 90° and 180°) and two equal acute angles, with the two sides forming the obtuse angle equal in length. This article covers the definition, why such a triangle is possible, its properties, the area and perimeter formulas, six worked examples, and the common mistakes.

Isosceles Acute Triangle: Properties & Examples

An isosceles acute triangle has two equal sides, two equal angles, and all three angles less than 90°. This article covers the definition, why both labels can hold at once, its properties, the area and perimeter formulas, six worked examples, and the common mistakes students make.

Isosceles Right Triangle: Formulas & Examples

An isosceles right triangle is a right triangle whose two legs are equal, giving angles of 45°, 45°, and 90° and a fixed side ratio of 1 : 1 : √2. This article covers its definition, the 45-45-90 angle structure, area ($\tfrac{x^2}{2}$) and hypotenuse ($x\sqrt{2}$) formulas, six worked examples, and the mistakes students make most.

Isosceles Triangle - Definition, Types, Formulas

An isosceles triangle is a triangle with two sides of equal length — called the legs — and one side of different length called the base. The two angles opposite the equal sides (the base angles) are also equal.

Types of Triangles — Classification Matrix

Triangles are classified two ways — by **side lengths** (equilateral, isosceles, scalene) and by **angle measures** (acute, right, obtuse). Combining the two axes gives a $3 \times 3$ matrix with **seven** valid types and two impossible ones. This article gives the complete matrix, properties of each type, three worked examples, and the impossibilities that come from the triangle angle-sum theorem.

Consecutive Interior Angles: Theorem & Examples

Consecutive interior angles are the two non-adjacent interior angles that sit on the same side of a transversal, and the Consecutive Interior Angles Theorem says they are supplementary (sum to 180°) exactly when the two lines are parallel. This article covers the definition, the theorem, its converse for proving lines parallel, consecutive angles in a parallelogram, and six worked examples.

Same Side Interior Angles: Theorem & Examples

Same side interior angles are the two angles that lie between two lines and on the same side of the transversal, and when the two lines are parallel they are supplementary (their sum is 180°). This article covers the definition, the same side interior angles theorem and its converse, why they are supplementary, and six worked examples that solve for the supplementary angle.

Alternate Exterior Angles: Theorem & Examples

Alternate exterior angles are the pair of angles that lie outside two lines and on opposite sides of the transversal crossing them. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, and six worked examples.

Alternate Interior Angles: Theorem & Examples

Alternate interior angles are the pair of angles that sit between two lines and on opposite sides of the line crossing them, called a transversal. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, co-interior angles, and six worked examples.

Corresponding Angles — Postulate, Pair Table, Examples

Corresponding angles are pairs of angles that occupy the same relative position at each of the two intersections formed when a transversal crosses two lines. When the two lines are parallel, corresponding angles are equal.

Transversal — All 8 Angles and Pair Relationships

A transversal is a line that crosses two or more other lines at distinct points. When the transversal crosses two parallel lines, exactly $8$ angles form — grouped into four named pair-relationships (corresponding, alternate interior, alternate exterior, co-interior).

Angle Between Two Vectors: Formula & Examples

The angle between two vectors is found from their dot product: $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}$, so $\theta = \cos^{-1}!\left(\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}\right)$. This article covers the formula, where it comes from, 2D and 3D worked examples, the cross-product alternative, six examples, and the common mistakes.

Angle Bisector: Properties & Construction

An angle bisector is a ray, line, or segment that divides an angle into two equal smaller angles. This article covers the definition, the key properties, the compass-and-straightedge construction, the angle bisector of a triangle and the incenter, and six worked examples.

Angle Addition Postulate: Formula & Examples

The angle addition postulate states that if a point $B$ lies in the interior of $\angle AOC$, then the two smaller angles add to the whole: $\angle AOB + \angle BOC = \angle AOC$. This article covers the definition, the formula, how to use it to solve for an unknown angle or for $x$, the link to straight and right angles, and six worked examples.

Congruent Angles: Definition, Theorems, Examples

Congruent angles are two or more angles that have the same measure, written $\angle A \cong \angle B$ using the congruence symbol ≅. This article covers the definition and symbol, the theorems that guarantee congruent angles, the compass-and-straightedge construction, and six worked examples.

Linear Pair of Angles: Definition & Axiom

A linear pair of angles is two adjacent angles whose non-common sides form a straight line, so they always add to 180°. This article covers the definition, the linear pair axiom and its converse, how a linear pair differs from supplementary and vertical angles, and six worked examples.

Supplementary vs Complementary Angles — A Side-by-Side Comparison

Supplementary angles add to $180°$; complementary angles add to $90°$. This article compares the two side by side, gives three worked examples (Quick, Standard, Stretch), explains the most common mistake, and offers the mnemonic that stops students mixing them up — Complementary forms a Corner, Supplementary forms a Straight line.

Supplementary Angles — Definition, Properties, Examples

Supplementary angles are any two angles whose measures sum to exactly 180°. The two angles can be next to each other (forming a straight line — a linear pair) or completely separate — what matters is the sum, not the position.

Complementary Angles — Definition, Properties, Examples

Complementary angles are any two angles whose measures sum to exactly 90°. The two angles can sit side by side (forming a right angle — a corner) or be drawn anywhere on the page — what matters is the sum. This article covers the definition, properties, the two types (adjacent vs non-adjacent), the right-triangle connection, three worked examples.

Vertical Angles — Definition, Theorem, Proof, and Examples

Vertical angles (also called vertically opposite angles) are the pair of non-adjacent angles formed when two straight lines cross at a single point. Sitting opposite each other across the intersection, they share only a vertex — never a side. The Vertical Angles Theorem states that vertical angles are always congruent (equal in measure), no matter how the two lines are oriented.

Adjacent Angles — Definition, Properties, and Examples

Adjacent angles are two angles that share a common vertex, share a common side (arm), and do not overlap. They sit next to each other — the word adjacent comes from Latin adjacens, meaning "lying near". Adjacent angles can be any size; they don't have to add to a specific number.

What is Adjacent? Meaning, Adjacent Angles, Solved Examples

In geometry, adjacent means "next to each other" — sharing a common side, edge, or vertex. Adjacent angles share a vertex and a side but don't overlap. Adjacent sides in a polygon share a common vertex. Adjacent in a triangle (with respect to an angle) is the side touching the angle that isn't the hypotenuse.

Vertex Angle: Definition, Formula, Examples

The vertex angle is the angle formed between the two equal sides of an isosceles triangle, sitting opposite the base, while the two angles at the ends of the base are the equal base angles. This article covers the definition, the apex-versus-base distinction, the formula $\text{vertex} = 180^\circ - 2(\text{base angle})$, the polygon vertex angle, six worked examples, and the common mistakes.

90 Degree Angle: Definition & Construction

A 90 degree angle is a right angle: the exact quarter turn formed when two lines meet perpendicularly, marked with a small square instead of an arc. This article covers the definition, how to construct one with a compass and verify it with the 3-4-5 rule, where right angles hold up buildings, and six worked examples.

45 Degree Angle: Definition & Construction

A 45 degree angle is an acute angle that measures exactly half of a right angle (90° ÷ 2 = 45°), and it is the angle each leg makes with the hypotenuse in an isosceles right triangle. This article covers the definition, how to construct one with a compass and by paper folding, its trigonometric values, where it shows up, and six worked examples.

Types of Angles: Acute, Right, Obtuse, Reflex

The six types of angles by measure are acute (under 90°), right (exactly 90°), obtuse (90° to 180°), straight (180°), reflex (180° to 360°), and full (360°). This article defines each type with a diagram, then covers the angle-pair relationships — complementary, supplementary, adjacent, and vertical — and the common mistakes.

Plane Definition in Math: Meaning & Examples

In math, a plane is a perfectly flat, two-dimensional surface that extends infinitely in every direction and has no thickness. This article covers the plane definition in math, its properties, how three non-collinear points fix exactly one plane, how planes are named, the difference between parallel and intersecting planes, and six worked examples.

Collinear Points: Definition, How to Prove & Examples

Collinear points are three or more points that all lie on the same straight line. This article covers the definition, collinear versus non-collinear points, the three methods to prove collinearity (slope equality, zero triangle area, and the distance test), how each one works, and six worked examples.

Coplanar: Definition, Points, Lines & Examples

Coplanar means lying on the same flat plane: a set of points is coplanar if a single plane can contain all of them, and lines are coplanar if one plane holds both. This article covers the definition, coplanar versus collinear, coplanar and non-coplanar points and lines, how to test for coplanarity, and six worked examples.

Skew Lines: Definition, Distance & Examples

Skew lines are two straight lines in three-dimensional space that never intersect and are never parallel, because they lie in different planes. This article covers the definition, why skew lines exist only in 3D, how to spot them in a cube, the conditions that classify two lines, the distance formula, and six worked examples.

Congruent Lines: Definition, Properties & Examples

Congruent lines are really line segments of equal length, written $\overline{AB} \cong \overline{CD}$ using the congruence symbol $\cong$. This article explains why true lines cannot be congruent, the three properties of congruent segments, how to check congruence with a ruler or the distance formula, and six worked examples.

Segment Bisector: Definition, Types & Examples

A segment bisector is any point, line, ray, or segment that passes through the midpoint of a line segment and splits it into two equal halves. This article covers the definition, the four kinds of bisector, how the perpendicular bisector is the special $90°$ case, the midpoint formula for finding one, and six worked examples.

Perpendicular Bisector: Definition & Construction

A perpendicular bisector of a line segment is a line that cuts the segment into two equal halves and meets it at a right angle ($90°$). Every point on it is equidistant from the two endpoints. This article covers the definition, the properties, the equidistance theorem, the compass-and-straightedge construction, the circumcentre, and six worked examples.

Perpendicular Lines - Definition, Slope, Examples

Line Segment: Definition, Properties & Examples

A line segment is a part of a straight line bounded by two distinct endpoints, so it has a fixed, measurable length — unlike a line or a ray, which run on forever. This article covers the definition and notation, how a segment differs from a line and a ray, its properties, the distance formula for length, and six worked examples.

Lines in Geometry: Types & Examples

Lines in geometry are straight, one-dimensional paths that have no thickness and run on forever in both directions. This article sorts out the point–line–ray–segment family first, then walks through every type a student meets: horizontal, vertical, parallel, perpendicular, intersecting, transversal, and skew lines, each with a labelled diagram and worked examples.

Geometry — Concepts, Formulas, Shapes & Topics Guide

Geometry is the branch of mathematics that studies points, lines, angles, shapes, and space — how figures are built, measured, and related. This hub maps every geometry topic into ten clusters: foundations (points, lines, planes), angles, parallel lines and transversals, triangles, quadrilaterals and polygons, circles, 3D solids, coordinate geometry, conic sections, and transformations — each linking to a full guide with formulas and worked examples.