Pythagoras’ Theorem
6.1 Pythagoras’ theorem – Cheat Sheet
Theorem Statement
In a right-angled triangle:
a² + b² = c²
Where:
- a, b = lengths of the shorter sides (legs)
- c = length of the hypotenuse (longest side, opposite the right angle)
Formulas Table
| Find | Formula | Notes |
|---|---|---|
| Hypotenuse c | c = √(a² + b²) | Use when both legs are known |
| Shorter side a | a = √(c² – b²) | Use when hypotenuse and other leg are known |
| Shorter side b | b = √(c² – a²) | Same as above, sides swapped |
Key Points
- Only applies to right-angled triangles
- The hypotenuse is always opposite the right angle
- All lengths must be in the same unit before calculation
- Can be extended to 3D problems (e.g., space diagonals)
Worked Examples
| Example | Given | Workings | Answer |
|---|---|---|---|
| 1 | a = 3 cm, b = 4 cm | c = √(3² + 4²) = √(9 + 16) = √25 | c = 5 cm |
| 2 | c = 13 m, b = 5 m | a = √(13² – 5²) = √(169 – 25) = √144 | a = 12 m |
| 3 (3D) | Cube with edge 6 cm, find space diagonal | Face diagonal = √(6² + 6²) = √72. Space diagonal = √((√72)² + 6²) = √(72 + 36) = √108 | ≈ 10.39 cm |