G5GeometryFoundation & Higher

Pythagoras' Theorem

Pythagoras' theorem connects the three sides of a right-angled triangle. If you know two sides, you can always find the third — a hugely useful tool in geometry and real life.

40 min Video by Maths Genie AQA GCSE Maths

Pythagoras and TrigonometryWatch the walkthrough, then read the notes below.

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What you'll learn

State Pythagoras' theorem
Find the hypotenuse
Find a shorter side
Use it in problems and on coordinate grids

The theorem

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

a^{2} + b^{2} = c^{2}

c is the hypotenuse (opposite the right angle).

c^{2} = 3^{2} + 4^{2} = 9 + 16 = 25

c = 25 = 5

cm.

Answer5 cm

Finding a shorter side

To find a shorter side, rearrange: $a^{2} = c^{2} - b^{2}$ . Subtract (not add) when the hypotenuse is one of the sides you already know.

Tip — The hypotenuse is always the longest side and is opposite the right angle.

Remember these

a^{2} + b^{2} = c^{2}

Finding the hypotenuse.

a^{2} = c^{2} - b^{2}

Finding a shorter side.

Watch out for these

Adding the squares when finding a shorter side.

Subtract: a² = c² − b².

Forgetting to square root at the end.

You find the square of the side first, then square root.

Key takeaways

a² + b² = c² for right-angled triangles.
Hypotenuse = longest side, opposite the right angle.
Add to find the hypotenuse; subtract to find a shorter side.

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