The theorem
In any right triangle:
a² + b² = c²
where a and b are the legs (the sides forming the right angle) and c is the hypotenuse (the longest side, opposite the right angle).
Named after the Greek mathematician Pythagoras, this relationship was also known to the ancient Babylonians, Egyptians, Indians and Chinese.
Seeing why it works
Draw a square on each side of a right triangle. The area of the big square on the hypotenuse equals the two smaller squares combined.
For a 3–4–5 triangle: 9 + 16 = 25 ✓
Finding the hypotenuse
Example: legs 6 and 8
- c² = 36 + 64 = 100
- c = 10
Finding a leg
Example: hypotenuse 13, one leg 5
- a² + 25 = 169
- a² = 144 → a = 12
When finding a leg, subtract.
Pythagorean triples
Whole-number solutions are called Pythagorean triples:
| 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 |
Multiples work too: 6, 8, 10 and 9, 12, 15.
The converse
If the sides of a triangle satisfy a² + b² = c², the triangle is a right triangle. This converse lets builders check square corners: measure 3 m along one wall, 4 m along the other; if the diagonal is 5 m, the corner is square.
| Compare c² with a² + b² | Triangle type |
|---|---|
| c² = a² + b² | right |
| c² < a² + b² | acute |
| c² > a² + b² | obtuse |
Applications
Ladder: A 5 m ladder reaches 4.8 m up a wall. How far is the base from the wall?
b² = 25 − 23.04 = 1.96 → b = 1.4 m
TV size: A TV screen is 120 cm wide and 68 cm tall. Diagonal?
√(14,400 + 4,624) = √19,024 ≈ 138 cm (about 54 inches)
Distance on a map: Walk 300 m east and 400 m north. Straight-line distance: 500 m.
3D diagonal of a box: d = √(l² + w² + h²). For 3 × 4 × 12: √(9 + 16 + 144) = 13.
Worked examples
Example 1: Legs 9 and 12. Hypotenuse? 15.
Example 2: Hypotenuse 25, leg 7. Other leg? √(625 − 49) = 24.
Example 3: Is a triangle with sides 6, 7, 9 right? 36 + 49 = 85 ≠ 81 → no (acute).
Common mistakes
- Adding when finding a leg.
- Using a side that isn’t the hypotenuse as c.
- Forgetting to take the square root at the end.
- Using the theorem on non-right triangles.
Try it yourself
- Legs 8 and 15. Hypotenuse?
- Hypotenuse 10, leg 6. Other leg?
- Is 20, 21, 29 a Pythagorean triple?
- A rectangle is 9 m by 40 m. Diagonal?
- Legs 2 and 3. Hypotenuse (exact)?
Answers: 1) 17 2) 8 3) yes 4) 41 m 5) √13
Quick recap
- Pythagorean theorem: a² + b² = c² in a right triangle.
- c is the hypotenuse; subtract to find a leg.
- The converse tests whether a corner is square.