Hypotenuse Calculator | Pythagorean Theorem
Find the hypotenuse or a missing leg of a right triangle with the Pythagorean theorem. Shows every step, the simplified root and the decimal answer.
- Hypotenuse
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- Simplified root
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Step by step
c = √(a² + b²)
This calculator is free, fast and ad-free. No sign-up, and what you type in isn't sent to us.
How the calculator works
The Pythagorean theorem links the three sides of a right triangle:
a² + b² = c²
The legs a and b are the two sides that meet at the right angle. The hypotenuse c is the side opposite the right angle, and it is always the longest of the three.
The calculator has two modes. Hypotenuse c takes the two legs and finds c = √(a² + b²). Missing leg b takes the hypotenuse and the leg you know and finds b = √(c² − a²). Under the answer you get every step: both squares, the sum or the difference, and the square root. An answer shown with = is exact. An answer shown with ≈ is rounded to four decimal places.
Finding the hypotenuse
Take a right triangle with legs of 3 in and 4 in:
- a² = 3² = 9
- b² = 4² = 16
- c² = 9 + 16 = 25
- c = √25 = 5
The hypotenuse is 5 in. Because 25 is a perfect square, the answer is exact.
Finding a leg
When you know the hypotenuse and one leg, you subtract instead of add. With c = 13 and a = 5:
- c² = 13² = 169
- a² = 5² = 25
- b² = 169 − 25 = 144
- b = √144 = 12
The missing leg is 12. If the hypotenuse you enter is not longer than the leg, the calculator shows an error, because no right triangle has sides like that.
Simplified root
Most pairs of legs do not give a whole number. With two legs of 5, c² = 25 + 25 = 50, so c = √50.
Since 50 = 25 × 2 and 25 is a perfect square, √50 = √25 × √2 = 5√2 ≈ 7.0711. The calculator shows √50 = 5√2 in its own row: the root is exact, the decimal is rounded. Decimals work the same way. Legs of 0.1 and 0.2 give c² = 0.05 and c = √5/10 ≈ 0.2236.
Pythagorean triples
A Pythagorean triple is three whole numbers that fit a² + b² = c². A right triangle with these sides has a whole-number hypotenuse, which makes the triples handy for checking a calculation. Five examples:
| a | b | c | a² + b² = c² |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 6 | 8 | 10 | 36 + 64 = 100 |
| 9 | 12 | 15 | 81 + 144 = 225 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
(6, 8, 10) is (3, 4, 5) with every side doubled, and (9, 12, 15) is (3, 4, 5) times 3.
Common mistakes
- Putting the hypotenuse in the wrong place. The hypotenuse is always the longest side, opposite the right angle. If you enter 5 and 13 as the two legs, you get √194 ≈ 13.9284 instead of the missing leg 12.
- Mixing units. The unit is only a label, and the calculator converts nothing. A leg of 9 in and a leg of 1 ft must be in the same unit first: 9 in and 12 in give 15 in. Entering 9 and 1 gives √82 ≈ 9.0554, which is neither inches nor feet.
- Using the theorem on a triangle without a right angle. a² + b² = c² holds only when the angle between a and b is exactly 90°. A triangle with sides 4, 5 and 6 has no right angle: legs of 4 and 5 would need a hypotenuse of √41 ≈ 6.4031.
Frequently asked questions
How do I find a leg when I know the hypotenuse?
Use b = √(c² − a²). With a hypotenuse of 10 and a leg of 6, b² = 100 − 36 = 64 and b = 8. In the calculator, choose Missing leg b.
Why does the answer sometimes show ≈?
When c² or b² is not a perfect square, the root has no exact decimal form. The calculator then rounds to four decimal places, and a very small answer gets four significant digits instead. Legs of 1 and 1 give c = √2 ≈ 1.4142.
Can I enter decimals?
Yes. Legs of 0.3 and 0.4 give 0.09 + 0.16 = 0.25, so the hypotenuse is exactly 0.5. The squares and the sum are worked out from the digits you type.
Does the calculator convert between units?
No. The unit is a label, and the answer is in the unit you enter. Enter both sides in the same unit.