Pythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle, with its area, perimeter and angles.
Missing side
5
- Area
- 6
- Perimeter
- 12
- Angle opposite leg a
- 36.87°
- Angle opposite leg b
- 53.13°
The hypotenuse is 5.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| Hypotenuse | √(3² + 4²) | 5 |
How it works
Choose whether to find the hypotenuse or a missing leg, then enter the sides you know.
Worked example
A right triangle with legs 3 and 4:
- Hypotenuse: √(3² + 4²) = √25 = 5
- Area: 3 × 4 ÷ 2 = 6. Perimeter: 3 + 4 + 5 = 12
- Angles: arctan(3 ÷ 4) = 36.87° and 53.13°
Sources
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 two legs, a² + b² = c². The hypotenuse is the longest side, opposite the right angle.
How do you find the hypotenuse?
Square both legs, add them and take the square root. For legs of 3 and 4, 9 + 16 = 25 and √25 = 5.
How do you find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root. With a hypotenuse of 13 and a leg of 5, 169 − 25 = 144 and √144 = 12.
Why must the hypotenuse be longer than the leg?
The hypotenuse is always the longest side of a right triangle. If the leg you enter is as long as or longer than the hypotenuse, no right triangle exists, so the calculator shows an error.
What are Pythagorean triples?
They are sets of whole numbers that fit the theorem, such as 3-4-5, 5-12-13 and 8-15-17. Any multiple of a triple also works, like 6-8-10.