Pythagoras' Theorem

When to use it

Right-angled triangles, to find a missing side (no angles needed).

The theorem

a² + b² = c²

where c is the hypotenuse — the longest side, always opposite the right angle.

Finding the hypotenuse (ADD)

Sides 3 cm and 4 cm:

c² = 3² + 4² = 9 + 16 = 25
c  = √25 = 5 cm

Finding a shorter side (SUBTRACT)

Hypotenuse 13 cm, one side 5 cm:

b² = 13² − 5² = 169 − 25 = 144
b  = √144 = 12 cm

Common Pythagorean triples

Worth recognising: 3,4,5 · 5,12,13 · 8,15,17 · 6,8,10.

In 3D

The diagonal of a cuboid: d = √(l² + w² + h²).

Distance between two points

distance = √[(x₂−x₁)² + (y₂−y₁)²] — Pythagoras on a coordinate grid.

Exam tip

Add for the hypotenuse, subtract for a shorter side. Identify the hypotenuse first (opposite the right angle). Leave answers as a surd, e.g. √20 = 2√5, if told to.