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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.

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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

  1. c² = 6² + 8² = 36 + 64 = 100
  2. 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

  1. a² = 13² − 5² = 169 − 25 = 144
  2. 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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