Pythagoras' Theorem
AQA Mathematicsfoundation⭐ High PriorityYears 9-10
What You Need to Know
- Only works for RIGHT-ANGLED triangles
- a² + b² = c² where c is the hypotenuse (longest side, opposite the right angle)
- Can be rearranged to find any missing side
- Used in 3D problems too — look for right angles in 3D shapes
Finding the Hypotenuse
When you know the two shorter sides, use c² = a² + b² to find the hypotenuse.
Worked Example
A right-angled triangle has sides 5cm and 12cm. Find the hypotenuse.
c² = a² + b²c² = 5² + 12² = 25 + 144 = 169c = √169 = 13
Answer: 13cm
Finding a Shorter Side
When you know the hypotenuse and one side, rearrange to a² = c² - b².
Worked Example
A right-angled triangle has hypotenuse 15cm and one side 9cm. Find the other side.
a² = c² - b²a² = 15² - 9² = 225 - 81 = 144a = √144 = 12
Answer: 12cm
Common Mistakes
- Using Pythagoras on non-right-angled triangles — check for the right angle first!
- Forgetting to square root at the end — you get c², not c
- Subtracting instead of adding (or vice versa) — remember c is always the hypotenuse
Exam Tips
- Draw the triangle and label sides before calculating
- In 3D problems, identify which face contains the right angle
- Give answers to 3 significant figures unless told otherwise
Related Topics
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