The Pythagorean theorem is one of the oldest and most widely used relationships in all of mathematics. Over 3,500 years old, it bridges ancient geometry with modern technology — from building pyramids to rendering 3D graphics and navigating with GPS. And it all starts with one simple equation: **a² + b² = c²**.
What does a² + b² = c² actually mean?
In any right triangle (a triangle with one **90-degree angle**), the two shorter sides are called **legs** (a and b), and the longest side opposite the right angle is the **hypotenuse** (c). The theorem says that the square of the hypotenuse equals the sum of the squares of the two legs. Picture a square built on each side of the triangle, and the area of the square on the hypotenuse equals the combined area of the squares on the two legs.
That geometric insight is the core of the theorem. You don't need to be a mathematician to use it — that's the beauty of it.
How do you use the theorem to solve for any side?
The formula rearranges depending on what you need. To find the hypotenuse: **c = √(a² + b²)**. To find a missing leg: **a = √(c² - b²)** or **b = √(c² - a²)**.
For the classic 3-4-5 triangle, a = 3 and b = 4, so c = √(9 + 16) = √25 = **5**. The Right Triangle Calculator on this site also uses SOH CAH TOA trigonometry to find the two acute angles of the triangle — not just the side lengths — giving you a complete picture of every measurement.
The theorem only works for right triangles. For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(90°) = 0, and the Law of Cosines simplifies right back to the Pythagorean theorem.
What are Pythagorean triples and why do they matter?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c² exactly. The most famous is **3-4-5**: 9 + 16 = 25. Any multiple of a triple is also a triple: 6-8-10, 9-12-15, 12-16-20 all work.
These triples were known to Babylonian mathematicians over a thousand years before Pythagoras was born, and they appear on ancient clay tablets. There are infinitely many triples, and Euclid described a formula to generate them.
| Triple | a² + b² | c² | Common Use |
|---|---|---|---|
| 3-4-5 | 9 + 16 = 25 | 25 | Squaring corners in construction |
| 5-12-13 | 25 + 144 = 169 | 169 | Navigation and surveying |
| 8-15-17 | 64 + 225 = 289 | 289 | Engineering calculations |
| 7-24-25 | 49 + 576 = 625 | 625 | Architecture and design |
Solve any right triangle instantly
Use our Right Triangle Calculator to find the hypotenuse, legs, area, perimeter, and angles — all with step-by-step SOH CAH TOA trigonometry, plus a dynamic SVG diagram.