GCSE Maths Pythagoras' Theorem Notes
Pythagoras' theorem links the three sides of a right-angled triangle: a² + b² = c². Use it to find any missing side when you know the other two.
Key concepts
- Hypotenuse
- The longest side, opposite the right angle. It is c in a² + b² = c².
- Finding the hypotenuse
- Add the squares of the two shorter sides, then square root.
- Finding a shorter side
- Subtract the squares (c² − a²), then square root.
Key formulas
Pythagoras' theorem
a² + b² = c²
Missing shorter side
a = √(c² − b²)
Worked example
A ladder 13 m long leans so its base is 5 m from a wall. How high up the wall does it reach?
- 1The ladder is the hypotenuse (13 m); the base distance is 5 m.
- 2Rearrange: height² = 13² − 5² = 169 − 25 = 144.
- 3Square root: height = √144 = 12.
Answer: 12 m
Common mistakes
✗ Adding the squares when finding a shorter side.
✓ To find a shorter side, subtract: a² = c² − b².
✗ Forgetting to square root at the end.
✓ You find the square of the side first, so always take the root.
✗ Treating a non-right-angled triangle with Pythagoras.
✓ Only use it when the triangle has a right angle.
Quick check
Try these, then reveal the answers.
Legs 3 and 4 — find the hypotenuse.Reveal answer
5
Hypotenuse 10, one leg 6 — find the other leg.Reveal answer
8
Which side is 'c' in a² + b² = c²?Reveal answer
The hypotenuse.
Exam tips
- Sketch the triangle and label the hypotenuse first.
- Decide whether you are finding the longest side (add) or a shorter side (subtract).
- Keep answers exact as surds when told, e.g. √20 = 2√5.
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Frequently asked questions
How do I know which side is the hypotenuse?+
It is the longest side and is always opposite the right angle.
Can Pythagoras find an angle?+
No — it only relates the sides. Use trigonometry (SOH CAH TOA) to find angles.
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