Pythagoras' Theorem, Explained

Over 2000 years ago, the ancient Greeks worked out a rule connecting the three sides of any right-angled triangle. It's still one of the most useful facts in maths — used constantly for measuring distances you can't measure directly.

Naming the sides

A right-angled triangle has one 90° corner. The side opposite that corner — always the longest side — is the hypotenuse. The other two sides are the shorter legs.

The theorem

Pythagoras' theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides:

a² + b² = c²

where c is the hypotenuse, and a and b are the two legs.

Finding the hypotenuse

Say the two shorter sides are 8 and 15.

Step 1 — square both legs and add them:

8² + 15² = 64 + 225 = 289

Step 2 — take the square root:

c = √289 = 17

So the hypotenuse is 17.

Finding a shorter side

The formula rearranges just as easily when it's a leg you're missing instead. Say the hypotenuse is 10 and one leg is 6.

Step 1 — square the hypotenuse and the known leg:

10² − 6² = 100 − 36 = 64

Step 2 — take the square root:

b = √64 = 8

So the missing leg is 8. Notice the only difference from finding the hypotenuse is that you subtract instead of add — you're always isolating the unknown square before rooting it.

Triples worth recognising

Some right-angled triangles have whole-number sides that come up again and again: 3-4-5, 6-8-10, 8-15-17, and 5-12-13. Spotting one of these instantly can save you the calculation.

In short

• The hypotenuse is the longest side, opposite the right angle.
• a² + b² = c².
• Add the squares of the legs to find the hypotenuse; subtract to find a missing leg.

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