Geometry formulas

Area, perimeter, volume and surface area for common 2D and 3D shapes, plus Pythagoras and Heron's formula.

These pages explain formulas for reference. They don't mean Mathover can solve every problem that uses them — interactive calculations are limited to the linked tools.

Area of a rectangle

A = l × w

Count how many unit squares fit: length times width.

Symbols

A: area
l: length
w: width

When it applies

l and w in the same unit; the area is in that unit squared.

Worked example

A rectangle is 8 cm by 3.5 cm. Find its area.

1.A = 8 × 3.5
2.A = 28

Answer: 28 cm²

Try it in the Rectangle Area & Perimeter tool →

Perimeter of a rectangle

P = 2(l + w)

The distance all the way round: two lengths plus two widths.

Symbols

P: perimeter
l: length
w: width

When it applies

l and w in the same unit; perimeter is in that unit (not squared).

Worked example

Find the perimeter of an 8 cm by 3.5 cm rectangle.

1.l + w = 8 + 3.5 = 11.5
2.P = 2 × 11.5 = 23

Answer: 23 cm

Try it in the Rectangle Area & Perimeter tool →

Area and perimeter of a square

A = s², P = 4s

A square is a rectangle with all four sides equal, so length × width becomes s × s.

Symbols

s: side length
A: area
P: perimeter

When it applies

s > 0. Area uses squared units; perimeter uses plain units.

Worked example

A square has side 6 m. Find its area and perimeter.

1.A = 6² = 36
2.P = 4 × 6 = 24

Answer: A = 36 m², P = 24 m

Try it in the Square Area & Perimeter tool →

Area of a circle

A = πr²

Square the radius and multiply by π (≈ 3.14159).

Symbols

A: area
r: radius (half the diameter)
π: pi ≈ 3.14159

When it applies

Use the radius, not the diameter. Leave the answer in terms of π for an exact value.

Worked example

Find the area of a circle with radius 5 cm.

1.r² = 5² = 25
2.A = 25π (exact)
3.25 × 3.14159… ≈ 78.54

Answer: 25π cm² ≈ 78.54 cm²

Try it in the Circle Area & Circumference tool →

Circumference of a circle

C = 2πr = πd

The distance around a circle is π times its diameter.

Symbols

C: circumference
r: radius
d: diameter = 2r

When it applies

Any circle. Units are plain length units.

Worked example

Find the circumference of a circle with radius 5 cm.

1.C = 2 × π × 5 = 10π (exact)
2.10 × 3.14159… ≈ 31.42

Answer: 10π cm ≈ 31.42 cm

Try it in the Circle Area & Circumference tool →

Area of a triangle

A = ½ × b × h

A triangle is half of a parallelogram with the same base and height.

Symbols

b: base
h: perpendicular height to that base

When it applies

h must be measured at right angles to the base — not a slanted side.

Worked example

A triangle has base 10 m and perpendicular height 6 m. Find its area.

1.A = ½ × 10 × 6
2.A = 60 ÷ 2 = 30

Answer: 30 m²

Try it in the Triangle Area & Perimeter tool →

Perimeter of a triangle and the triangle inequality

P = a + b + c, where a + b > c, a + c > b, b + c > a

Add the three sides. The sides only make a real triangle if every pair adds to more than the third.

Symbols

a, b, c: the three side lengths
P: perimeter

When it applies

All sides positive and satisfying the triangle inequality.

Worked example

A triangle has sides 5 cm, 7 cm and 9 cm. Check it exists and find its perimeter.

1.5 + 7 = 12 > 9, 5 + 9 = 14 > 7, 7 + 9 = 16 > 5 ✓
2.P = 5 + 7 + 9 = 21

Answer: 21 cm

Try it in the Triangle Area & Perimeter tool →

Heron's formula

A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2

Finds a triangle's area from its three sides alone, using the semi-perimeter s.

Symbols

a, b, c: side lengths
s: semi-perimeter
A: area

When it applies

The sides must form a valid triangle (triangle inequality).

Worked example

Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.

1.s = (13 + 14 + 15) ÷ 2 = 21
2.s − a = 8, s − b = 7, s − c = 6
3.21 × 8 × 7 × 6 = 7056
4.√7056 = 84

Answer: 84 cm²

Reference only — no interactive Mathover tool for this formula yet.

Pythagoras' theorem

a² + b² = c²

In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides.

Symbols

a, b: the two shorter sides (legs)
c: the hypotenuse, opposite the right angle

When it applies

Right-angled triangles only.

Worked example

A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.

1.c² = 6² + 8² = 36 + 64 = 100
2.c = √100 = 10

Answer: 10 cm

Try it in the Trigonometry Calculator tool →

Area of a trapezium (trapezoid)

A = ½(a + b)h

Average the two parallel sides, then multiply by the distance between them.

Symbols

a, b: the parallel sides
h: perpendicular distance between them

When it applies

Quadrilaterals with one pair of parallel sides.

Worked example

A trapezium has parallel sides 6 cm and 10 cm, 4 cm apart. Find its area.

1.a + b = 16
2.½ × 16 × 4 = 32

Answer: 32 cm²

Reference only — no interactive Mathover tool for this formula yet.

Volume of a cuboid

V = l × w × h

Area of the base times the height.

Symbols

l, w, h: length, width, height
V: volume

When it applies

All three in the same unit; the volume is in that unit cubed.

Worked example

Find the volume of a 5 cm × 4 cm × 3 cm box.

1.5 × 4 = 20
2.20 × 3 = 60

Answer: 60 cm³

Reference only — no interactive Mathover tool for this formula yet.

Surface area of a cuboid

SA = 2(lw + lh + wh)

A cuboid has three pairs of identical rectangular faces. Add one of each, then double.

Symbols

l, w, h: length, width, height
SA: total surface area

When it applies

Closed cuboids. Units are squared.

Worked example

Find the surface area of a 5 cm × 4 cm × 3 cm box.

1.lw = 20, lh = 15, wh = 12
2.20 + 15 + 12 = 47
3.2 × 47 = 94

Answer: 94 cm²

Reference only — no interactive Mathover tool for this formula yet.

Volume of a cylinder

V = πr²h

Area of the circular base times the height.

Symbols

r: radius of the base
h: height
V: volume

When it applies

Right circular cylinders. Units are cubed.

Worked example

A cylinder has radius 3 cm and height 10 cm. Find its volume.

1.r² = 9
2.V = π × 9 × 10 = 90π (exact)
3.90 × 3.14159… ≈ 282.74

Answer: 90π cm³ ≈ 282.74 cm³

Reference only — no interactive Mathover tool for this formula yet.

Surface area of a cylinder

SA = 2πr² + 2πrh

Two circular ends plus the curved side, which unrolls into a rectangle 2πr long and h tall.

Symbols

r: radius
h: height

When it applies

Closed right circular cylinders. Units are squared.

Worked example

Find the total surface area of a cylinder with radius 3 cm and height 10 cm.

1.Ends: 2π × 3² = 18π
2.Curved side: 2π × 3 × 10 = 60π
3.18π + 60π = 78π (exact)
4.78 × 3.14159… ≈ 245.04

Answer: 78π cm² ≈ 245.04 cm²

Reference only — no interactive Mathover tool for this formula yet.

Volume of a sphere

V = (4/3)πr³

Cube the radius, multiply by π and by 4/3.

Symbols

r: radius
V: volume

When it applies

Spheres. Units are cubed.

Worked example

Find the volume of a sphere with radius 3 cm.

1.r³ = 27
2.(4/3) × 27 = 36
3.V = 36π (exact) ≈ 113.10

Answer: 36π cm³ ≈ 113.10 cm³

Reference only — no interactive Mathover tool for this formula yet.

Surface area of a sphere

SA = 4πr²

A sphere's surface area is exactly four times the area of a circle with the same radius.

Symbols

r: radius

When it applies

Spheres. Units are squared.

Worked example

Find the surface area of a sphere with radius 2 m.

1.r² = 4
2.4 × 4 = 16
3.SA = 16π (exact) ≈ 50.27

Answer: 16π m² ≈ 50.27 m²

Reference only — no interactive Mathover tool for this formula yet.

Volume of a cone

V = (1/3)πr²h

A cone holds exactly one third of the cylinder with the same base and height.

Symbols

r: base radius
h: perpendicular height

When it applies

Right circular cones; h is the vertical height, not the slant height.

Worked example

A cone has radius 3 cm and height 4 cm. Find its volume.

1.r² = 9
2.(1/3) × 9 × 4 = 12
3.V = 12π (exact) ≈ 37.70

Answer: 12π cm³ ≈ 37.70 cm³

Reference only — no interactive Mathover tool for this formula yet.