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Pythagorean Theorem Calculator

This Pythagorean theorem calculator solves for the hypotenuse or a missing leg of a right triangle with worked steps. It also returns area, perimeter, and the two acute angles.

One side of the right angle. Leave blank to solve for a.

The other side of the right angle. Leave blank to solve for b.

The side opposite the right angle, and the longest side. Leave blank to solve for c.

Triangle

Hypotenuse c (solved)
5
Leg a
3
Leg b
4
Hypotenuse c
5
Area
6
Perimeter
12
Angle A (opposite a)
36.87°
Angle B (opposite b)
53.13°

3² + 4² = 5²

  1. The missing side is the hypotenuse c.
  2. Square the legs: 3² = 9 and 4² = 16.
  3. Add the squares: 9 + 16 = 25.
  4. Take the square root: c = √25 = 5.

Angle A is opposite leg a, and angle B is opposite leg b. The right angle is 90°. This applies only to right triangles.

How to use

  1. Enter any two sides: legs a and b, and hypotenuse c. Leave the missing side blank.
  2. Read the solved side and the steps. They use a² + b² = c².
  3. Check the area, the perimeter, and the two acute angles. The right angle stays 90°.
  4. If a leg is longer than or equal to the hypotenuse, those sides cannot form a right triangle.

Formula

In a right triangle, a² + b² = c², where c is the hypotenuse. The hypotenuse is c = √(a² + b²). A missing leg is a = √(c² − b²), and that difference must be positive. Area = (a × b) ÷ 2. Perimeter = a + b + c. The acute angles are arctan(a/b) and arctan(b/a), and they add to 90°.

Example. Legs of 3 and 4 give c = √(3² + 4²) = √(9 + 16) = √25 = 5. The area is (3 × 4) ÷ 2 = 6, the perimeter is 12, and the acute angles are 36.87° and 53.13°.

Example. With hypotenuse 13 and leg 5, the other leg is √(13² − 5²) = √(169 − 25) = √144 = 12.

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 legs: a² + b² = c².

How do I find a missing leg?

Rearrange to a = √(c² − b²). With c = 13 and b = 5, a = √(169 − 25) = 12.

Does it work for any triangle?

No. It only applies to right triangles. For other triangles use the law of cosines.

What is a Pythagorean triple?

A set of whole numbers that fit the theorem, like 3-4-5, 5-12-13, and 8-15-17.

How do builders use it?

The 3-4-5 rule checks a square corner: measure 3 ft on one side and 4 ft on the other; the diagonal should be exactly 5 ft.

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