What is the Pythagorean theorem?

In a right triangle, a² + b² = c², where c is the hypotenuse.

How do I find the hypotenuse?

Square the two shorter sides, add them, and take the square root.

Does it work for any triangle?

No, only right-angled triangles.

The Pythagorean Theorem Explained

Published 9/25/2025 · 2 min read · Everyday calculators

Lena Hoffmann — Science & education writer at OneKitly

Mathematics · Physics

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In short

The Pythagorean theorem says that in a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two: a² + b² = c². So a triangle with sides 3 and 4 has a hypotenuse of √(9+16) = √25 = 5. It only works for right triangles, and lets you find any side when you know the other two.

In a right triangle, a² + b² = c². Here's what the formula means, a worked example, how to find any side, and where it's used in real life.

The formula

The theorem links the three sides of a right-angled triangle — one with a perfect 90° corner. Label the two shorter sides (the legs) a and b, and the longest side (the hypotenuse, always opposite the right angle) c. The rule is a² + b² = c²: square each leg, add them, and you have the square of the hypotenuse. It's one of the oldest and most useful facts in all of geometry.

A worked example

Take a triangle whose legs are 3 and 4. Square them — 9 and 16 — and add to get 25, which is c². The hypotenuse is the square root of 25, so c = 5. This '3-4-5' triangle is the most famous example because the sides come out as whole numbers. Another common one is 5-12-13: 25 + 144 = 169, and √169 = 13. These tidy sets are called Pythagorean triples.

Finding a leg, not just the hypotenuse

The theorem works backwards too. If you know the hypotenuse and one leg, rearrange to b² = c² − a²: subtract instead of add, then take the square root. Say a ladder 5 m long leans with its foot 3 m from a wall; the height it reaches is √(25 − 9) = √16 = 4 m. Whenever you have any two of the three sides, this rearranged form finds the missing one.

Where it's used

Far from an abstract classroom fact, this theorem is everywhere. Builders use the 3-4-5 trick to check that a corner is a true right angle. It gives the straight-line distance between two points from their horizontal and vertical gaps — the basis of the distance formula and of navigation and computer graphics. It even explains screen sizes: a '27-inch' monitor is the diagonal, worked out from its width and height with exactly this rule.

Worked with our own calculator

Pythagorean theorem calculator

Given

Side a: 3
Side b: 4

Result

Hypotenuse (c): 5
Triangle area: 6

These figures are produced by the calculator below, not typed in by hand — they are recomputed whenever the tool changes.

Run it on your own figures →

Frequently asked questions

What is the Pythagorean theorem?: In a right triangle, a² + b² = c², where c is the hypotenuse.
How do I find the hypotenuse?: Square the two shorter sides, add them, and take the square root.
Does it work for any triangle?: No, only right-angled triangles.

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Sources

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