Pythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle with a² + b² = c², check whether three sides make a right triangle, and see the area, angles and steps
Last updated: November 9, 2026
How to Use the Pythagorean Theorem Calculator
How to use the Pythagorean theorem calculator
- Choose what to find. Pick the hypotenuse, a missing leg, or the check that tells you whether three sides form a right triangle.
- Enter the sides you know. The fields change with your choice. Use the same unit for every side, and add the unit at the bottom if you want it shown in the answer.
- Read the result. You get the missing side, the working step by step, and, for right triangles, the area, perimeter, both acute angles, the height to the hypotenuse and whether the sides form a Pythagorean triple.
When you solve for a leg, the hypotenuse must be longer than the leg you enter, because it is always the longest side. In the check mode you can type the three sides in any order, since the calculator treats the longest one as the hypotenuse.
The theorem only applies to right triangles. For other triangles you need the law of sines or the law of cosines.
Formula & Methodology
The Pythagorean theorem
In any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, called the legs:
a² + b² = c²
Rearranging it gives the three ways to find a missing side:
- Hypotenuse: c = √(a² + b²)
- Leg a: a = √(c² − b²)
- Leg b: b = √(c² − a²)
Worked examples
Finding the hypotenuse. For legs of 6 and 8: c = √(6² + 8²) = √(36 + 64) = √100 = 10.
Finding a missing leg. For a hypotenuse of 13 and one leg of 5: b = √(13² − 5²) = √(169 − 25) = √144 = 12.
A result that is not a whole number. For two legs of 1: c = √(1 + 1) = √2 ≈ 1.4142.
Pythagorean triples
A Pythagorean triple is a set of three whole numbers that fit the theorem exactly. The best-known examples are:
- 3, 4, 5 (and its multiples, such as 6, 8, 10 and 9, 12, 15)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- 20, 21, 29
Checking for a right angle
The theorem works in reverse too. If the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. If they add up to more, the triangle is acute, and if they add up to less, it is obtuse. Carpenters use the 3-4-5 rule to square a corner: mark 3 feet along one wall and 4 feet along the other, and the diagonal between the marks should measure exactly 5 feet.
Other measurements of a right triangle
- Area = (a × b) ÷ 2
- Perimeter = a + b + c
- Acute angles: one is tan⁻¹(a ÷ b), and the other is 90° minus that angle
- Height to the hypotenuse = (a × b) ÷ c
About Pythagorean Theorem Calculator
Our pythagorean theorem calculator is a free, easy-to-use tool designed to help youfind the hypotenuse or a missing leg of a right triangle with a² + b² = c², check whether three sides make a right triangle, and see the area, angles and steps. If you're a professional or just need quick calculations, our calculator provides accurate results instantly.
Frequently Asked Questions
How do you find the hypotenuse of a right triangle?
Square both legs, add the results, then take the square root: c = √(a² + b²). For legs of 3 and 4, that is √(9 + 16) = √25 = 5.
How do you find a missing side of a right triangle?
If the missing side is a leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root. For a hypotenuse of 10 and a leg of 6, the other leg is √(100 − 36) = √64 = 8.
Does the Pythagorean theorem work for any triangle?
No. It only works for right triangles, which have one 90° angle. For other triangles, use the law of cosines, which reduces to the Pythagorean theorem when the angle is 90°.
What is a Pythagorean triple?
It is a set of three positive whole numbers, such as 3, 4, 5 or 5, 12, 13, where a² + b² = c². Multiplying every number in a triple by the same whole number gives another triple, so 6, 8, 10 also works.
Which side is the hypotenuse?
The hypotenuse is the longest side of a right triangle, and it sits opposite the right angle. The two shorter sides that form the right angle are called the legs, and they can be labeled a and b in either order.
What is the 3-4-5 rule used for?
Builders and carpenters use it to make a square corner. Measure 3 units along one edge and 4 units along the other. If the distance between the two points is exactly 5 units, the corner is a true 90° angle. Any multiple works too, such as 6, 8 and 10 feet.
What units can I use?
Any unit of length works, as long as every side uses the same one. The answer is in that same unit, and the area is in square units of it. Mixing inches and feet in one calculation will give the wrong result.
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