How to Find the Hypotenuse of a Right Triangle (Step-by-Step)

To find the hypotenuse of a right triangle, use the Pythagorean Theorem: c = √(a² + b²), where a and b are the two legs and c is the hypotenuse (the side opposite the 90° angle). Square each leg, add the results, then take the square root.

For a triangle with legs 3 and 4, the hypotenuse is √(9 + 16) = √25 = 5. That's the whole idea — the rest is careful arithmetic and identifying which side is the hypotenuse.

Worked example: legs of 6 and 8

Find the hypotenuse of a right triangle whose legs measure 6 and 8.

1. Write the Pythagorean Theorem: a² + b² = c².
2. Substitute the legs: 6² + 8² = c².
3. Square each term: 36 + 64 = c².
4. Add: 100 = c².
5. Take the positive square root: c = √100 = 10.

The hypotenuse is 10. (Notice this is just the 3-4-5 triple doubled to 6-8-10.)

The Pythagorean Theorem, in plain steps

1. Identify the hypotenuse. It is always the side opposite the right angle, and it is always the longest side. Example: in a triangle with sides 5, 12, and 13, the 13 is the hypotenuse.
2. Label the two legs a and b. These are the sides that form the right angle. Example: legs a = 5 and b = 12.
3. Square both legs. Compute a² and b² separately. Example: 5² = 25 and 12² = 144.
4. Add the squares. That sum equals c². Example: 25 + 144 = 169.
5. Take the square root. c = √(a² + b²). Example: c = √169 = 13.

When you already know a Pythagorean triple

If the two legs match a known triple, you can skip the arithmetic. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, plus any multiple of those (like 6-8-10 or 9-12-15).

Second worked example: legs of 9 and 12

Find the hypotenuse of a right triangle with legs 9 and 12.

1. Set up: 9² + 12² = c².
2. Square: 81 + 144 = c².
3. Add: 225 = c².
4. Square root: c = √225 = 15.

The hypotenuse is 15. This is the 3-4-5 triple scaled up by 3 (giving 9-12-15), so spotting the pattern would have gotten you the answer instantly.

What if the legs don't make a clean number?

Sometimes c² won't be a perfect square. Legs of 2 and 3 give c² = 4 + 9 = 13, so c = √13 ≈ 3.61. Leave the answer in radical form (√13) when the problem asks for exact values, and use a decimal only if the problem asks you to approximate.

Common mistakes students make

• Adding the legs before squaring. (a + b)² is not a² + b². You must square first, then add.
• Using the theorem when the triangle isn't a right triangle. The Pythagorean Theorem only works when one angle is exactly 90°.
• Mislabeling the hypotenuse. If a problem gives you a hypotenuse and one leg and asks for the other leg, you'd use b² = c² − a², not a² + b².
• Forgetting to take the square root. Stopping at c² = 100 and writing 100 as the answer is the most common careless error.
• Losing the units. If the legs are in feet, the hypotenuse is in feet — not square feet.

The formula is short, but building fluency takes reps on different setups: legs given, hypotenuse given, word problems with ladders and diagonals, and triangles hiding inside squares or rectangles. LernOS's guided tutor walks you through each variation one question at a time, so you catch your own reasoning gaps instead of just copying steps. Sign up below to try it on your next problem.