Pythagoras' Theorem
Pythagoras' Theorem
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
Where:
- c is the hypotenuse — the side opposite the right angle, always the longest side
- a and b are the two shorter sides (called "legs")
This relationship was known to ancient Babylonians and Egyptians, but the Greek mathematician Pythagoras gave it the formal proof that made it famous.
Key Terms
- Hypotenuse: the longest side of a right-angled triangle, opposite the 90° angle
- Legs: the two shorter sides that form the right angle
- Pythagorean triple: a set of three whole numbers that satisfy a² + b² = c²
- Right angle: an angle of exactly 90°, marked with a small square in diagrams
Finding the Hypotenuse
When you know both shorter sides, substitute into c² = a² + b² and take the square root.
Example: Sides = 3cm and 4cm
- c² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5cm
Example: Sides = 8cm and 15cm
- c² = 8² + 15² = 64 + 225 = 289
- c = √289 = 17cm
Finding a Shorter Side
Rearrange the formula: b² = c² - a²
Always subtract the square of the known leg from the square of the hypotenuse.
Example: Hypotenuse = 13cm, one side = 5cm
- b² = 13² - 5² = 169 - 25 = 144
- b = √144 = 12cm
Example: Hypotenuse = 10cm, one side = 6cm
- b² = 10² - 6² = 100 - 36 = 64
- b = √64 = 8cm
Pythagorean Triples
These are sets of whole numbers that satisfy a² + b² = c² exactly — no messy square roots.
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 ✓ |
| 5, 12, 13 | 25 + 144 = 169 ✓ |
| 8, 15, 17 | 64 + 225 = 289 ✓ |
| 7, 24, 25 | 49 + 576 = 625 ✓ |
Any multiple of a Pythagorean triple also works:
- 6, 8, 10 (×2 of 3,4,5)
- 10, 24, 26 (×2 of 5,12,13)
- 9, 12, 15 (×3 of 3,4,5)
Recognising common triples saves significant time in exams.
Real-World Application — Ladder Problem
A ladder 10m long leans against a wall. Its foot is 6m from the wall. How high up the wall does it reach?
- The ladder = hypotenuse = 10m
- Distance from wall = one leg = 6m
- Height = b, the unknown leg
- b² = 10² - 6² = 100 - 36 = 64
- b = 8m
Diagonal of a Square
A square with side length s has a diagonal d.
Using Pythagoras (the diagonal is the hypotenuse of a right-angled triangle with both legs = s): d² = s² + s² = 2s² d = s × √2
Example: A square has sides of 5cm. Diagonal = 5√2 ≈ 7.07cm
3D Pythagoras
To find a diagonal distance in a 3D shape, apply the theorem twice.
Find the space diagonal of a cuboid 3cm × 4cm × 12cm:
- Step 1: Find the base diagonal: d² = 3² + 4² = 25, so d = 5cm
- Step 2: Use d and the height: space diagonal² = 5² + 12² = 25 + 144 = 169
- Space diagonal = 13cm
Checking if a Triangle is Right-Angled
If you are given three side lengths and want to check whether the triangle is right-angled, test whether a² + b² = c² where c is the longest side.
Example: sides 6, 8, 10 6² + 8² = 36 + 64 = 100 = 10² ✓ → right-angled triangle confirmed
Example: sides 4, 5, 7 4² + 5² = 16 + 25 = 41 ≠ 7² = 49 → not right-angled
Common Mistakes to Avoid
- Forgetting which side is the hypotenuse: it is always opposite the right angle and always the longest side
- Subtracting when finding the hypotenuse: use c² = a² + b² (add), not subtract
- Adding when finding a shorter side: use b² = c² - a² (subtract)
- Forgetting to square root at the end: c² = 25 means c = 5, not 25