Pythagorean Theorem Calculator
Find the missing side of a right triangle from the other two, with the exact square-root answer, the working, both angles, the area and the perimeter.
In a right triangle the two shorter sides, the legs a and b, and the longest side, the hypotenuse c, always satisfy a² + b² = c². Pick the side you want, enter the other two, and the calculator squares, adds or subtracts and takes the square root. Legs of 6 and 9 give c² = 36 + 81 = 117, so the hypotenuse is √117, which simplifies to 3√13, about 10.81665383.
Results assume the triangle has an exact right angle and that both sides are measured in the same unit. Decimals are rounded for display; the radical form is exact. For construction or engineering work, measure carefully and allow for the precision of your tools.
The legs a and b meet at the right angle; the hypotenuse c is the longest side, opposite it. Any unit works, as long as both sides use the same one. Sides up to 1,000,000.
Hypotenuse c
10.81665383= 3√13
- c² = a² + b²
- c² = 6² + 9² = 36 + 81 = 117
- c = √117 = 3√13 ≈ 10.81665383
- Angle opposite a
- 33.69°
- Angle opposite b
- 56.31°
- Area
- 27
- Perimeter
- 25.81665383
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About the Pythagorean Theorem Calculator
The theorem is Euclid's Proposition 47 in Book I of the Elements: the square on the side opposite the right angle equals the two squares on the sides that form it. To find the hypotenuse the calculator adds the squares of the legs and takes the square root; to find a leg it subtracts the square of the known leg from the square of the hypotenuse, so the hypotenuse has to be the longer of the two you enter. When both sides you enter are whole numbers, the answer is also shown in simplest radical form (for very large sides only the decimal is shown), found by pulling every pair of equal prime factors out from under the root, so √117 becomes 3√13 because 117 is 9 times 13. The decimal is the correctly rounded square root, shown to ten significant digits, and when all three sides are whole numbers the triangle is a Pythagorean triple, like the 3, 4 and 5 triangle often used to check that a corner is square, or the 5, 12 and 13 triangle. The two acute angles come from the arctangent of one leg over the other and always add up to 90 degrees. Use the same unit for both sides and the answer comes back in it.
Frequently asked questions
How do I find the hypotenuse?+
Square each leg, add the squares and take the square root. With legs of 3 and 4 that is 9 plus 16, which is 25, and the square root of 25 is 5. Choose Hypotenuse c in the Find box, enter the two legs and the working appears line by line.
How do I find a missing leg?+
Square the hypotenuse, subtract the square of the leg you know and take the square root. With a hypotenuse of 17 and a leg of 8, that is 289 minus 64, which leaves 225, and its square root is 15. The leg you enter must be shorter than the hypotenuse, since no right triangle has a leg as long as its longest side.
What does 3√13 mean?+
It is an exact way to write a square root that does not come out whole. Since 117 is 9 times 13 and the square root of 9 is 3, the root of 117 equals 3 times the root of 13. Many textbooks ask for answers in this form, and the decimal next to it is the same number rounded.
Does it work for triangles without a right angle?+
No. The theorem holds only when one angle is exactly 90 degrees. For other triangles the side lengths are linked by the law of cosines instead, which adds a term for the angle between the two known sides.
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