What is important to know about Pythagoras' theorem?

In a right-angled triangle: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle).

What is important to know about Pythagoras' theorem?

To find the hypotenuse: c = √(a² + b²).

What is important to know about Pythagoras' theorem?

To find a shorter side: a = √(c² − b²).

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Geometry & Measures · Foundation & Higher

Pythagoras' theorem

Pythagoras' theorem relates the three sides of a right-angled triangle. It only works in right-angled triangles.

Revising pythagoras' theorem the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Pythagoras' theorem GCSE” on the site for the complete guide.

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Key facts to remember

1In a right-angled triangle: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle).
2To find the hypotenuse: c = √(a² + b²).
3To find a shorter side: a = √(c² − b²).
4Check: the hypotenuse is always opposite the right angle and always the longest side.
5Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 (scale these up too: 6-8-10, etc.).

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Formulas

Pythagoras' theorem

a² + b² = c²

c is the hypotenuse

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Worked examples

Example 1

Find the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm.

Working

c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10

Answer10 cm

Example 2

A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.

Working

a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12

Answer12 cm

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Common mistakes

✗Squaring and adding all three sides instead of squaring the two legs.

✗Not identifying which side is the hypotenuse before applying the formula.

✗Forgetting to take the square root at the end.

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Exam tips

✓Label the hypotenuse as c first — then identify which sides you know and which you need.

✓Pythagorean triples save time on non-calculator papers — memorise 3-4-5 and 5-12-13.

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