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Maths · Geometry & Measures

Pythagoras' theorem

Find the missing side in right-angled triangles.

Not startedCAMBRIDGE · 8CBSE · VIIIIB · MYP-2

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What is Pythagoras' theorem?

Pythagoras' theorem is a rule about right-angled triangles. It says the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. We write it as a² + b² = c², where c is the hypotenuse.

Finding the hypotenuse

When you know the two shorter sides, add their squares together. Then take the square root to find the hypotenuse. For example, if the sides are 3 and 4, then 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

Finding a shorter side

When you know the hypotenuse and one shorter side, subtract the square of the shorter side from the hypotenuse squared. Then take the square root. If c = 10 and a = 6, then b² = 10² − 6² = 100 − 36 = 64, so b = 8.

The right-angle matters

Pythagoras' theorem only works for right-angled triangles. The hypotenuse is always opposite the right angle and is the longest side. Make sure you identify which side is which before calculating.

Checking your answer

Always check if your answer makes sense. The hypotenuse must be longer than either of the other two sides. If your calculation gives a smaller number, check your working carefully.

Worked examples

  1. 1. A right-angled triangle has sides 5 cm and 12 cm. Find the hypotenuse.

    1. Use a² + b² = c² where a = 5 and b = 12.
    2. Calculate: 5² + 12² = 25 + 144 = 169.
    3. Find the square root: c = √169 = 13 cm.

  2. 2. The hypotenuse is 13 m and one side is 5 m. Find the other side.

    1. Use a² + b² = c² and rearrange to b² = c² − a².
    2. Substitute: b² = 13² − 5² = 169 − 25 = 144.
    3. Find the square root: b = √144 = 12 m.

  3. 3. A triangle has sides 8 cm and 15 cm meeting at a right angle. Find the hypotenuse.

    1. Apply a² + b² = c² with a = 8 and b = 15.
    2. Calculate: 8² + 15² = 64 + 225 = 289.
    3. Take the square root: c = √289 = 17 cm.

Try it yourself

A right-angled triangle has sides 9 cm and 12 cm. What is the hypotenuse?

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