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Geometry formulas

The measurement formulas of school geometry: Pythagoras' theorem, the areas of rectangles, triangles, parallelograms, trapezoids, circles, sectors and ellipses, the angles and areas of polygons, and the volumes and surface areas of boxes, cylinders, cones, spheres and pyramids. Each one solves backwards too: give the area and get the radius, give the volume and get the height.

Printable cheat sheet Formula finder

Pythagorean theorem

a² + b² = c²

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Area of a rectangle

A = l×w

Length times width: the number of unit squares that fit inside a rectangle.

Area of a triangle

A = 1/2bh

Half the base times the perpendicular height, for any triangle.

Heron's formula

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

The area of a triangle from its three side lengths alone, with no height or angle needed.

Area of a parallelogram

A = bh

Base times perpendicular height; the slanted sides do not matter.

Area of a trapezoid (trapezium)

A = (a + b)/2h

The average of the two parallel sides times the distance between them.

Area of a circle

A = πr²

Pi times the radius squared: the space inside a circle.

Circumference of a circle

C = 2πr

The distance around a circle: two pi times the radius, or pi times the diameter.

Arc length

s = rθ (θ in radians), s = θ/360×2πr (θ in degrees)

The length of part of a circle's edge, cut off by a central angle θ.

Area of a sector

A = 1/2r²θ (θ in radians), A = θ/360πr² (θ in degrees)

The area of a pizza-slice shaped part of a circle with central angle θ.

Area of an ellipse

A = πab

Pi times the two semi-axes: a circle's area formula, stretched.

Area of a regular polygon

A = ns²/(4tan (π/n))

The area of any regular polygon (equal sides and equal angles) from the number of sides and the side length.

Interior angles of a polygon

S = (n − 2)×180^°

The interior angles of any polygon with n sides add up to (n − 2) × 180°.

Volume of a box (cuboid)

V = lwh

Length times width times height: the space inside a rectangular box.

Surface area of a box (cuboid)

S = 2(lw + lh + wh)

The total area of the six rectangular faces of a box.

Volume of a cylinder

V = πr²h

The area of the circular base times the height.

Surface area of a cylinder

S = 2πr² + 2πrh

Two circular ends plus the curved side, which unrolls into a rectangle.

Volume of a cone

V = 1/3πr²h

One third of the cylinder with the same base and height.

Surface area of a cone

S = πr² + πrl

The circular base plus the curved surface, using the slant height l.

Volume of a sphere

V = 4/3πr³

Four thirds pi r cubed: the space inside a ball.

Surface area of a sphere

S = 4πr²

Exactly four times the area of a circle with the same radius.

Volume of a pyramid

V = 1/3Bh

One third of the base area times the height, for a base of any shape.