Concepts · 3 min read
Pythagorean Theorem: A Visual Proof and Pythagorean Triples
A proof in four triangles
Take four identical right triangles with legs a and b and hypotenuse c. Arrange them inside a big square of side (a + b) so their hypotenuses form a tilted square of side c in the middle.
The big square has area (a + b)² = a² + 2ab + b². It is also the tilted square plus four triangles: c² + 4 × (½ab) = c² + 2ab. Setting the two equal and cancelling 2ab leaves a² + b² = c².
Check with a = 3, b = 4: (3 + 4)² = 49, and 5² + 2 × 3 × 4 = 49. ✓
Using it
Given two sides you can find the third. Try the Pythagorean theorem calculator.
Legs 5 and 12
- Area of the triangle: 30
- Perimeter: 30
Step-by-step working 4 steps
- 1The theorem
a² + b² = c²
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
- 2Square the legs and add
5² + 12² = 25 + 144 = 169
- 3Take the square root
c = √169 = 13
- 4Check
5² + 12² = 169 and 13² = 169
Pythagorean triples
Whole-number solutions are called triples. Euclid’s formula builds them: for m > k with opposite parity and no common factor, (m² − k², 2mk, m² + k²).
| a | b | c | a² + b² | c² |
|---|---|---|---|---|
| 3 | 4 | 5 | 25 | 25 |
| 5 | 12 | 13 | 169 | 169 |
| 8 | 15 | 17 | 289 | 289 |
| 7 | 24 | 25 | 625 | 625 |
| 20 | 21 | 29 | 841 | 841 |
| 12 | 35 | 37 | 1369 | 1369 |
| 9 | 40 | 41 | 1681 | 1681 |
| 28 | 45 | 53 | 2809 | 2809 |
| 11 | 60 | 61 | 3721 | 3721 |
| 33 | 56 | 65 | 4225 | 4225 |
The converse
If a² + b² = c² for three side lengths, the triangle must have a right angle — the reason builders check corners with a 3-4-5 loop of string. See the geometry cheat sheet for more formulas.
Frequently asked questions
Did Pythagoras discover it?
The relationship was known in Babylonian and Indian mathematics before Pythagoras; the theorem carries his name because of the Greek tradition of proving it.
Does it work in 3D?
Yes: the space diagonal of a box is √(l² + w² + h²).
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