How to Use the Pythagorean Theorem

The Pythagorean theorem applies to right triangles — triangles with one 90-degree angle. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².

It only works for right triangles, but within them it's exact and unfailing, which is why it underpins everything from geometry class to construction, navigation, and computer graphics.

To find the hypotenuse, square both legs, add them, and take the square root: c = √(a² + b²). With legs of 3 and 4, that's √(9 + 16) = √25 = 5 — the famous 3-4-5 triangle.

To find a missing leg when you know the hypotenuse, rearrange to b = √(c² − a²). With a hypotenuse of 10 and one leg of 6, the other leg is √(100 − 36) = √64 = 8.

If you know a side and an angle instead of two sides, trigonometry finishes the job: sine, cosine, and tangent relate the angle to the sides. The calculator handles every combination and shows the steps.

Some right triangles have whole-number sides, called Pythagorean triples, and memorizing a few speeds up mental math. The common ones are 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Any multiple of a triple is also a triple — 6-8-10 and 9-12-15 are just the 3-4-5 triangle scaled up. Builders use the 3-4-5 rule constantly to check that a corner is square.