# Pythagoras' theorem

In a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides: a^2 + b^2 = c^2.

Maths (school) · Part 06: Pythagoras and trigonometry · Card 1 of 9 · Last updated 2026-10-19

## Analogy

Like fitting square tiles onto each side: the two small squares cover exactly the big one.

## Example

**Find the hypotenuse**

Short sides 5 and 12
25 + 144 = 169
sqrt(169) = 13
To find a short side: 10^2 - 6^2 = 64, so 8

## In practice

Builders check a corner is square by measuring 3, 4 and 5 along the sides.

## Practice

### Flashcards

- **What are the short sides of a right triangle if the hypotenuse is 13 and one side is 5?** 12. 169 - 25 = 144, and sqrt(144) = 12.

### Interview questions

**Find the length of the hypotenuse of a right-angled triangle with shorter sides 5 cm and 12 cm. Explain each step.**

- Pythagoras: a^2 + b^2 = c^2
- 25 + 144 = 169
- c = sqrt(169)
- c = 13 cm

## Questions people ask

### Which side is the hypotenuse?

The hypotenuse is the longest side of a right-angled triangle. It is always opposite the right angle. In a^2 + b^2 = c^2, the letter c stands for the hypotenuse. The other two sides are interchangeable. Spot it first, before putting any numbers into the formula, to avoid mixing the sides up.

### When do you subtract in Pythagoras?

You subtract when you are finding one of the shorter sides and you know the hypotenuse. Rearranging gives a^2 = c^2 - b^2. For a hypotenuse of 10 and a side of 6, a^2 = 100 - 36 = 64, so a = 8.

## Related concepts

- [Distance between two points](https://studyvisually.com/learn/science/maths-school/maths-school-part-6/maths-school-part-6-concept-2)

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