Pythagoras' Theorem
Learn Pythagoras' Theorem (a² + b² = c²) with real-world examples — ladders, TV screens, and distances. Includes 3D Pythagoras and Pythagorean triples.
Pythagoras' Theorem is one of the most useful things in all of maths. It lets you find a missing side of a right-angled triangle when you know the other two.
The Formula
Where c is always the hypotenuse — the longest side, opposite the right angle. a and b are the other two sides.
Finding the Hypotenuse (Long Side)
Worked Example 1 — Finding the long side
A right-angled triangle has sides of 6 cm and 8 cm. Find the hypotenuse.
- c² = 6² + 8² = 36 + 64 = 100
- c = √100 = 10 cm
Finding a Shorter Side
Rearrange: a² = c² − b²
Worked Example 2 — Finding a short side
Hypotenuse = 13 cm, one side = 5 cm. Find the other side.
- a² = 13² − 5² = 169 − 25 = 144
- a = √144 = 12 cm
Real-World Examples
Worked Example 3 — Ladder Against a Wall
A 5-metre ladder leans against a wall. The bottom is 3 metres from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse (c = 5), the ground distance is one side (b = 3)
- a² = 5² − 3² = 25 − 9 = 16
- a = √16 = 4 metres
TV screen sizes are measured diagonally. A "55-inch" TV with a width of 48 inches has a height of √(55² − 48²) = √(3025 − 2304) = √721 ≈ 26.8 inches.
Pythagorean Triples
These are sets of whole numbers that satisfy a² + b² = c². They're handy to memorise because they come up a lot:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Any multiple also works — so 6, 8, 10 (double of 3,4,5) is also a Pythagorean triple.
3D Pythagoras (A-Level Preview)
To find the space diagonal of a cuboid (like the longest line you could fit inside a box):
Example: a box is 3m × 4m × 12m. The space diagonal = √(9 + 16 + 144) = √169 = 13m.
Try It Yourself
- Find the hypotenuse of a triangle with sides 9 cm and 12 cm. (Answer: 15 cm)
- A right triangle has hypotenuse 17 cm and one side 8 cm. Find the other. (Answer: 15 cm)
- Find the space diagonal of a cuboid 2m × 3m × 6m. (Answer: 7m)