Geometry Formulas: Area, Perimeter & Volume for 2D and 3D Shapes

Geometry formulas are the mathematical rules used to calculate the size and dimensions of shapes - their perimeter, area, surface area, and volume.

The formulas split into two groups:

1. 2D or plane geometry formulas, which apply to flat shapes like squares, triangles, and circles

2. 3D or solid geometry formulas, which apply to objects with depth, like cubes, cylinders, and spheres.

Every formula on this page uses a small set of variables (side, length, width, radius, height) that are defined in the variable key below.

Quick-Reference: Which Geometry Formula Do You Need?

Before picking a formula, identify what you are measuring. The table below maps the quantity to the formula family and the correct unit of the answer.

LSA is lateral surface area - the sides only, excluding the top and bottom. CSA is curved surface area, used when the surface is curved, as in a cylinder or cone. TSA is total surface area, which includes every face.

Variable Key: What the Letters in Geometry Formulas Mean

Every geometry formula is written using shorthand letters for the dimensions of a shape. The same letter can mean different things in different formulas, so the notation below is the one used throughout this article.

When a shape has more than one side of unequal length, extra letters appear - for example, a scalene triangle uses a, b, and c for its three sides. Where this happens, the formula below makes the variable meanings explicit.

Geometry Formulas for 2D Shapes

Two-dimensional shapes have length and width but no depth. The two quantities measured are perimeter (the distance around the shape) and area (the space inside it). The circle has its own term - circumference - for its perimeter.

These formulas appear in CCSS 6.G.1 and NCERT Class 6 Chapter 10.

Square

A square is a four-sided shape with all sides equal and all angles 90°.

• Perimeter: P = 4s

• Area: A = s²

Rectangle

A rectangle has opposite sides of equal length and all four angles 90°.

• Perimeter: P = 2(l + w)

• Area: A = l × w

• Diagonal: d = √(l² + w²)

Triangle (General)

A triangle has three sides and three angles that sum to 180°.

• Perimeter: P = a + b + c

• Area: A = ½ × b × h

When only the three side lengths are known and the height is not given, Heron's formula applies:

A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2.

Right Triangle

A right triangle has one 90° angle. The two sides forming the right angle are called legs; the side opposite the right angle is the hypotenuse.

• Perimeter: P = a + b + c

• Area: A = ½ × a × b (where a and b are the legs)

• Pythagoras theorem: a² + b² = c² (where c is the hypotenuse)

Equilateral Triangle

An equilateral triangle has all three sides equal and all three angles equal to 60°.

• Perimeter: P = 3s

• Area: A = (√3 / 4) × s²

Isosceles Triangle

An isosceles triangle has two equal sides (a) and a third side (b).

• Perimeter: P = 2a + b

• Area: A = ½ × b × h

Circle

A circle is the set of all points at a fixed distance from a centre point. The distance from the centre to the edge is the radius; edge to edge through the centre is the diameter.

• Circumference: C = 2πr or C = πd

• Area: A = πr²

The value of π is approximately 3.14159, and it is the same for every circle regardless of size.

Parallelogram

A parallelogram has two pairs of parallel sides. Opposite sides are equal in length.

• Perimeter: P = 2(a + b)

• Area: A = b × h

The height here is the perpendicular distance between the base and the opposite side — not the length of the slanted side.

Rhombus

A rhombus is a parallelogram with all four sides equal. Its diagonals meet at 90°.

• Perimeter: P = 4a

• Area: A = ½ × d₁ × d₂

Trapezium (Trapezoid)

A trapezium has one pair of parallel sides (a and b) and two non-parallel sides (c and d).

• Perimeter: P = a + b + c + d

• Area: A = ½ × (a + b) × h, where h is the perpendicular distance between the parallel sides

Kite

A kite has two pairs of equal-length sides that are adjacent rather than opposite. The diagonals meet at 90°.

• Perimeter: P = 2(a + b)

• Area: A = ½ × d₁ × d₂

Regular Polygon (n sides)

A regular polygon has n equal sides. The apothem is the perpendicular distance from the centre to any side.

• Perimeter: P = n × s

• Area: A = ½ × P × apothem

For a regular hexagon with side s, the area simplifies to A = (3√3 / 2) × s².

Geometry Formulas for 3D Shapes

Three-dimensional shapes have length, width, and height. The three quantities measured are lateral or curved surface area (the sides only), total surface area (all faces), and volume (the space enclosed).

These formulas appear in CCSS 7.G.6 and 8.G.9, and in NCERT Class 8 Chapter 11, Class 9 Chapter 13, and Class 10 Chapter 12.

Cube

A cube has six equal square faces and all edges the same length (a).

• LSA: 4a²

• TSA: 6a²

• Volume: a³

Cuboid (Rectangular Prism)

A cuboid has six rectangular faces, with length (l), width (w), and height (h).

• LSA: 2h(l + w)

• TSA: 2(lw + wh + lh)

• Volume: l × w × h

Cylinder

A cylinder has two parallel circular faces joined by a curved surface.

• CSA: 2πrh

• TSA: 2πr(r + h)

• Volume: πr²h

Cone

A cone has a circular base and narrows to a single point called the apex. The slant height (sₗ) is the distance from the apex to the edge of the base, given by sₗ = √(r² + h²).

• CSA: πr × sₗ

• TSA: πr(r + sₗ)

• Volume: ⅓ πr²h

Sphere

A sphere is the set of all points at a fixed distance from a centre point in 3D space. A sphere has no flat faces and no edges.

• Surface Area: 4πr²

• Volume: ⁴⁄₃ πr³

Hemisphere

A hemisphere is half of a sphere, formed by cutting a sphere along a plane through its centre.

• CSA: 2πr² (the curved part only)

• TSA: 3πr² (curved part plus the flat circular base)

• Volume: ⅔ πr³

Prism (General)

A prism is a 3D shape with two identical parallel bases and flat sides joining them. The base can be any polygon — triangle, rectangle, pentagon, hexagon.

• LSA: Perimeter of base × height

• TSA: LSA + 2 × (area of base)

• Volume: Area of base × height

Pyramid (General)

A pyramid has a polygon base and triangular faces that meet at a single apex. The slant height (sₗ) is the distance from the apex to the midpoint of a base edge.

• LSA: ½ × Perimeter of base × sₗ

• TSA: LSA + Area of base

• Volume: ⅓ × Area of base × height

Units in Geometry: A Quick Reminder

Geometry formulas produce answers in specific units depending on what is being measured. Using the wrong unit is one of the most common mistakes on school assessments.

• Perimeter and circumference use linear units: cm, m, inches, feet.

• Area and surface area use square units: cm², m², in², ft².

• Volume uses cubic units: cm³, m³, in³, ft³.

Converting between units requires raising the conversion factor to the appropriate power. 1 m = 100 cm, so 1 m² = 10,000 cm² (not 100 cm²), and 1 m³ = 1,000,000 cm³.

Worked Examples Using Geometry Formulas

Example 1: Area of a Rectangle

A rectangle has length 12 cm and width 7 cm. Find its area.

A = l × w A = 12 × 7 A = 84 cm²

Example 2: Surface Area and Volume of a Cylinder

A cylinder has radius 5 cm and height 10 cm. Find its total surface area and volume.

TSA = 2πr(r + h) TSA = 2π(5)(5 + 10) TSA = 2π(5)(15) TSA = 150π TSA ≈ 471.24 cm²

Volume = πr²h Volume = π(5²)(10) Volume = 250π Volume ≈ 785.40 cm³

Example 3: Area of an Equilateral Triangle

An equilateral triangle has side length 6 cm. Find its area.

A = (√3 / 4) × s² A = (√3 / 4) × 36 A = 9√3 A ≈ 15.59 cm²

Related Formulas and Common Variations

The table below lists formulas that often sit alongside the main geometry set. These appear most frequently in coordinate geometry, advanced mensuration, and higher-grade problems.

Geometry Formulas in School Curricula