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The Pythagorean Theorem: How to Find the Hypotenuse
In any right-angled triangle, a² + b² = c². Learn how the Pythagorean theorem works, how to find the hypotenuse or a missing leg, and where it shows up in everyday measuring.
Updated 2 November 2026 · 2 min read
The Pythagorean theorem is one of the most useful rules in all of maths, and it applies to every right-angled triangle — a triangle with one 90° corner. It links the lengths of the three sides so tightly that, knowing any two, you can always find the third.
The rule in one line
In words: square the two shorter sides, add them together, and the result equals the square of the longest side. The hypotenuse is always the longest side, and it always sits across from the 90° corner — a quick way to spot which side is which.
Finding the hypotenuse
Say a triangle has legs of 3 and 4. Then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. That neat 3-4-5 triangle is the most famous example, but the method works for any values. To get the hypotenuse you add the squares and then take a square root to undo the squaring.
Finding a missing leg
If you already know the hypotenuse and one leg, rearrange the formula to subtract instead of add. With a hypotenuse of 13 and one leg of 5: a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12. The right triangle calculator takes the two legs and returns the hypotenuse, the area, the perimeter and the other two angles, so you can check any of these by hand against it.
Where it's genuinely useful
- Checking that a corner is truly square when building a deck, wall or raised bed — the 3-4-5 trick confirms a right angle without a set square.
- Finding the straight-line (diagonal) distance across a rectangular room or garden.
- Working out the length of a ramp, rafter or ladder reaching a known height.
- Measuring the diagonal of a TV or monitor from its width and height.
The theorem only holds for right-angled triangles, so the first step is always to confirm there's a 90° corner. Once squaring and square-rooting feel comfortable, they open the door to a lot of geometry — the area of a circle is another formula built on the same squaring idea.
Frequently asked questions
- Which side is the hypotenuse?
- The hypotenuse is the side opposite the right angle, and it's always the longest of the three. The other two sides, which meet at the 90° corner, are called the legs.
- Does the theorem work for any triangle?
- No — only for right-angled triangles. If none of the angles is 90°, a² + b² = c² doesn't hold, and you'd need the law of cosines instead.
- What is a Pythagorean triple?
- A set of three whole numbers that fit the theorem exactly, such as 3-4-5, 5-12-13 and 8-15-17. They're handy because the hypotenuse comes out as a whole number with no messy decimals.