Pythagorean Theorem Calculator

Pythagorean theorem at a glance#

The Pythagorean theorem relates the three sides of a right triangle: a² + b² = c², where a and b are the two legs and c is the hypotenuse, the side opposite the right angle. To find the hypotenuse, square both legs, add them, then take the square root of the total.

With legs of 3 and 4, c² = 3² + 4² = 9 + 16 = 25, so the hypotenuse c is the square root of 25, which is 5. With legs of 6 and 8, c² = 36 + 64 = 100, so c = 10. To find a missing leg instead, rearrange to a² = c² minus b² and take the square root.

These four sets are common Pythagorean triples: whole numbers where a² + b² equals c² exactly, so 5² + 12² = 25 + 144 = 169 = 13². Enter any two sides in the calculator above for the exact third side, plus the area and perimeter. Most real measurements are not whole-number triples, so expect a decimal result that you round to suit your work.

Finding each side#

The theorem a² + b² = c² lets you solve for whichever side you are missing, as long as you know the other two. The legs a and b form the right angle; c is the hypotenuse opposite it.

Finding the hypotenuse#

Square both legs, add them, then take the square root: c = √(a² + b²). With legs 6 and 8, c = √(36 + 64) = √100 = 10.

Finding a leg#

Rearrange to a = √(c² − b²). If the hypotenuse is 13 and one leg is 5, the other leg is √(169 − 25) = √144 = 12.

Pythagorean triples#

A Pythagorean triple is three whole numbers where a² + b² = c² exactly, so the sides come out as integers instead of decimals. The 3-4-5 triple is the smallest: 9 + 16 = 25. Any multiple of a triple is also a triple, so 6-8-10 and 9-12-15 work too. Builders use 3-4-5 to check a corner is square without a protractor: mark 3 units along one edge and 4 along the other, and the diagonal should measure exactly 5.

Where you use it#

Use the theorem any time you need a straight-line distance across a right angle: the diagonal of a rectangular room, the length of a brace across a frame, or the distance between two points on a grid. Most real measurements are not clean triples, so expect a decimal result that you round to suit the job.

Frequently asked questions#

What is the Pythagorean theorem?#

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It only holds for triangles with a 90-degree angle.

How do I find the hypotenuse?#

Square both legs, add them, then take the square root: c = √(a² + b²). For legs 3 and 4, c = √(9 + 16) = √25 = 5.

Can I find a leg instead of the hypotenuse?#

Yes. Rearrange to a = √(c² − b²). Enter the hypotenuse and one leg, and the calculator returns the missing leg. With c = 10 and b = 6, the other leg is √(100 − 36) = √64 = 8.

What are common Pythagorean triples?#

3-4-5, 5-12-13, 8-15-17 and 7-24-25 are the most common. Each set satisfies a² + b² = c² with whole numbers, and any multiple of a triple is also a triple.

Does the theorem work for non-right triangles?#

No. For triangles without a 90-degree angle you need the law of cosines, c² = a² + b² − 2ab·cos(C). The Pythagorean theorem is the special case where angle C is 90 degrees and cos(C) is 0.