What is Pythagoras' theorem and how do you use it?
Short answer
In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². Use it to find a missing side when you know the other two.
Step by step
- Check the triangle has a right angle and find the hypotenuse, the side opposite it.
- To find the hypotenuse: square the two shorter sides, add them, then take the square root.
- To find a shorter side: square the hypotenuse, subtract the square of the known short side, then take the square root.
- Check your answer makes sense: the hypotenuse must be the longest side.
Worked example
A ladder 5 m long leans against a wall with its foot 1.4 m from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse: c = 5
- h² = 5² − 1.4² = 25 − 1.96 = 23.04
- h = √23.04 = 4.8 m
Common mistakes
- Adding the squares when you are looking for a shorter side. If you need a short side, subtract.
- Forgetting the final square root.
- Using the theorem on a triangle with no right angle.
Useful for:GCSEBECEBEPCCheckpoint
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