What Is A Quadrilateral?
A quadrilateral is a flat shape with four straight sides, four vertices, and four angles. The four interior angles always sum to 360°. The name comes from the Latin quadri (four) and latus (side), and any closed figure you can draw with four line segments, from a perfect square to a lopsided arrowhead, belongs to this family.
Two facts hold for every quadrilateral, no matter how stretched or skewed: it has exactly two diagonals (line segments joining opposite corners), and its angles total 360°. Everything else, such as equal sides, parallel sides, or right angles, is what splits the family into the named types below.
Why Does The Angle Sum Equal 360°?
Here is the cleanest way to see it. Draw any one diagonal of a quadrilateral, say from A to C, and that single line splits the four-sided shape into two triangles. The angles inside a triangle always add to 180°, a result that traces back to Euclid's Elements. Two triangles give $2 \times 180° = 360°$, and those angles are exactly the four corner angles of the quadrilateral.
$$\text{Angle sum} = 2 \times 180° = 360°$$
This is the engine behind the whole family. When a shape forces some angles to be 90°, the others are pinned down too, which is why a rectangle's "fourth" angle is never a free choice.
What Are The Six Types of Quadrilaterals?
There are six named quadrilaterals every student meets. The trick is to stop memorising them as six separate shapes and see them as a hierarchy — each one is a more constrained version of a looser one.
Parallelogram: both pairs of opposite sides are parallel and equal. Opposite angles are equal; the diagonals bisect each other.
Rectangle: a parallelogram with all four angles equal to 90°. Diagonals are equal and bisect each other.
Square: a rectangle with all four sides equal. It is the most constrained, with equal sides, right angles, and equal diagonals that meet at 90°.
Rhombus: a parallelogram with all four sides equal. Diagonals bisect each other at right angles, but the angles are not all 90° (unless it is a square).
Trapezium (called a trapezoid in US usage): exactly one pair of parallel sides. The parallel sides are the bases.
Kite: two pairs of adjacent equal sides. One diagonal bisects the other at a right angle.
A square satisfies every rule at once, which is why it sits at the bottom of the hierarchy: every square is a rectangle, every square is a rhombus, and every rectangle, rhombus, and square is a parallelogram. The reverse is never guaranteed: a rectangle is only a square when its sides happen to be equal.
How Are Quadrilaterals Classified (convex vs concave)?
Before the named types, quadrilaterals split into two broad groups by shape.
A convex quadrilateral has every interior angle less than 180°, and both diagonals sit inside the figure. All six named types above are convex. A concave quadrilateral (sometimes called a re-entrant or arrowhead quadrilateral) has one interior angle greater than 180°, and one diagonal falls outside the shape. There is also a third, rarer case, the complex or crossed quadrilateral, where two sides intersect, but it shows up far less often at this level.
A useful check: if you can connect every pair of corners with a diagonal that stays inside the shape, it is convex. If one diagonal escapes outside, it is concave.
Properties of Quadrilaterals at a Glance
Every comparison table below assumes standard labelling: opposite sides, the two diagonals $d_1$ and $d_2$, and the relevant side lengths.
Type | Sides | Angles | Diagonals |
|---|---|---|---|
Parallelogram | Opposite sides equal and parallel | Opposite angles equal | Bisect each other |
Rectangle | Opposite sides equal | All four are 90° | Equal, bisect each other |
Square | All four equal | All four are 90° | Equal, bisect at 90° |
Rhombus | All four equal | Opposite angles equal | Bisect each other at 90° |
Trapezium | One pair parallel | Co-interior angles on each leg sum to 180° | Not generally equal |
Kite | Two adjacent pairs equal | One pair of opposite angles equal | One bisects the other at 90° |
Area And Perimeter Formulas For Each Quadrilateral
The area formulas look unrelated until you notice they all come from the same idea: chop the shape into pieces you already know how to measure. Here is each formula with what every symbol means.
Square, side $s$: $$\text{Area} = s^2 \qquad \text{Perimeter} = 4s$$
Rectangle, length $l$ and width $w$: $$\text{Area} = l \times w \qquad \text{Perimeter} = 2(l + w)$$
Parallelogram, base $b$ and perpendicular height $h$: $$\text{Area} = b \times h$$
The height $h$ is the straight-line distance between the two parallel sides, not the slanted side. (Slide the triangle off one end of a parallelogram to the other and you get a rectangle of base $b$ and height $h$ — that rearrangement is where $b \times h$ comes from.)
Rhombus, diagonals $d_1$ and $d_2$: $$\text{Area} = \frac{1}{2} , d_1 d_2$$
Trapezium, parallel sides $a$ and $b$ with height $h$: $$\text{Area} = \frac{1}{2}(a + b),h$$
Kite, diagonals $d_1$ and $d_2$: $$\text{Area} = \frac{1}{2} , d_1 d_2$$
The rhombus and kite share a formula because both have diagonals that meet at a right angle, and any shape whose diagonals are perpendicular has area equal to half their product.
Examples of Quadrilaterals
Example 1
A quadrilateral has three angles measuring 80°, 95°, and 100°. Find the fourth angle.
The four angles sum to 360°.
$$80° + 95° + 100° = 275°$$ $$\text{Fourth angle} = 360° - 275°$$ $$\text{Fourth angle} = 85°$$
Final answer: 85°.
Example 2
Identify the most specific name for a quadrilateral with all four sides equal and all four angles equal to 90°.
First instinct: many readers stop at "rhombus" because all four sides are equal, and a rhombus is the all-sides-equal shape. Let us test it. A rhombus does have four equal sides, but its angles are not required to be 90°, whereas this shape has both equal sides and right angles. So "rhombus" is true but not the most specific name.
The shape that demands equal sides and four right angles is the square. A square is simultaneously a rhombus (all sides equal) and a rectangle (all angles 90°), so the most specific correct name is square.
Final answer: Square.
Example 3
Find the area of a parallelogram with base 12 cm and perpendicular height 7 cm.
$$\text{Area} = b \times h$$ $$\text{Area} = 12 \times 7$$ $$\text{Area} = 84 \text{ cm}^2$$
Final answer: 84 cm².
Example 4
A rhombus has diagonals of 10 cm and 24 cm. Find its area.
$$\text{Area} = \frac{1}{2} , d_1 d_2$$ $$\text{Area} = \frac{1}{2} \times 10 \times 24$$ $$\text{Area} = 120 \text{ cm}^2$$
Final answer: 120 cm².
Example 5
A trapezium has parallel sides of 8 m and 14 m, with a height of 5 m. Find its area.
$$\text{Area} = \frac{1}{2}(a + b),h$$ $$\text{Area} = \frac{1}{2}(8 + 14) \times 5$$ $$\text{Area} = \frac{1}{2} \times 22 \times 5$$ $$\text{Area} = 55 \text{ m}^2$$
Final answer: 55 m².
Example 6
A kite has diagonals measuring 16 cm and 9 cm. Find its area, then explain why the same formula works for a rhombus.
$$\text{Area} = \frac{1}{2} , d_1 d_2$$ $$\text{Area} = \frac{1}{2} \times 16 \times 9$$ $$\text{Area} = 72 \text{ cm}^2$$
The formula works for both because in a kite and a rhombus the diagonals cross at 90°. When two diagonals are perpendicular, they cut the shape into right triangles whose combined area always reduces to half the product of the diagonals.
Final answer: 72 cm².
Where The Quadrilateral Family Earns Its Keep
Quadrilaterals are not a syllabus invention. The classification exists because the rules a shape obeys decide what it can do. A rectangle is the default for a screen, a door, or a brick because right angles tile and stack without gaps. A parallelogram's defining property, that opposite sides stay parallel as it skews, is exactly what makes the pantograph and the parallel-motion linkage in drafting machines work: the moving arm stays parallel to the fixed one through the whole motion.
The hierarchy matters for the same reason. When an engineer proves a result about parallelograms (say, that the diagonals bisect each other), that proof is inherited for free by rectangles, rhombuses, and squares, because each is a parallelogram. You prove once at the top of the family and reuse all the way down.
Map software leans on the same idea: a region on a screen is stored as a polygon, and knowing whether four corner points form a convex quadrilateral decides whether a simple fill algorithm will work or whether the concave case needs special handling.
Tripping Points To Avoid
Mistake 1: Treating the types as six unrelated shapes
Where it slips in: When a question asks for the most specific name, or asks whether a square "counts as" a rectangle.
Don't do this: Picking the first name whose definition matches and stopping there, as if square, rectangle, and rhombus were rival shapes that exclude each other.
The correct way: Read the family as a hierarchy. A square satisfies the rectangle rules and the rhombus rules, so it sits below both. When two names both fit, the more constrained one (more equalities, more right angles) is the more specific answer.
Mistake 2: Confusing the slant side with the perpendicular height
Where it slips in: Area of a parallelogram or trapezium, where the figure is drawn tilted and a slanted side is the longest labelled length on the page.
Don't do this: Multiplying base by the slanted side. The first move students reach for is to grab the two numbers attached to the corner and multiply, which quietly substitutes the slant length for the height.
The correct way: The height is the perpendicular distance between the parallel sides, the dashed line that meets both at 90°. If the diagram gives only the slant side, you find the height first, often with the Pythagorean theorem, before using $\text{Area} = b \times h$.
Mistake 3: Forgetting the angle sum is fixed at 360°
Where it slips in: "Find the missing angle" problems, and proofs that need one more angle than the question states.
Don't do this: Assuming the four angles are independent and leaving the missing one blank because "there isn't enough information."
The correct way: Any three angles of a quadrilateral fix the fourth, because all four must total 360°. This is the most reliable single fact in the whole topic, and it is exactly the kind of step students skip because it feels too obvious to write down, only to lose the mark when the proof needed it stated.
Conclusion
A quadrilateral is any four-sided polygon, and its four interior angles always sum to 360°.
The six named types (parallelogram, rectangle, square, rhombus, trapezium, kite) form a hierarchy where each is a more constrained version of a looser one.
A square is the most specific quadrilateral: it is a rectangle and a rhombus and a parallelogram simultaneously.
Rhombus and kite share the area formula $\frac{1}{2} d_1 d_2$ because both have perpendicular diagonals.
The perpendicular height, not the slant side, drives every parallelogram and trapezium area calculation.
A Practical Next Step
Practice these problems to solidify your understanding, working from the angle-sum questions up to the area formulas.
A quadrilateral has angles of 110°, 70°, and 95°. Find the fourth angle. (Answer to Question 1: 85°)
Find the area of a rhombus whose diagonals are 12 cm and 16 cm. (Answer to Question 2: 96 cm²)
Name the most specific quadrilateral with all sides equal and no right angles. (Answer to Question 3: rhombus)
If you get stuck on the area questions, return to the formulas section and check whether you are using the perpendicular height.
Want a live Bhanzu trainer to walk through more quadrilateral problems and the reasoning behind each formula? Book a free demo class.
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