Pythagorean Theorem Calculator
Find the missing side of a right triangle using the Pythagorean theorem. Calculate the hypotenuse or a leg with our free online calculator.
How to Use This Calculator
- Choose Find Hypotenuse to calculate the hypotenuse from two legs, or Find Leg to calculate a missing leg from one leg and the hypotenuse.
- Enter the known values in the fields provided.
- Click Calculate to see the missing side, area, perimeter, and a visual diagram.
Formula
Finding hypotenuse: c = √(a² + b²)
Finding leg: a = √(c² − b²)
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Common Pythagorean Triples
| Triple | a | b | c (hyp) | Area |
|---|---|---|---|---|
| (3, 4, 5) | 3 | 4 | 5 | 6 |
| (5, 12, 13) | 5 | 12 | 13 | 30 |
| (8, 15, 17) | 8 | 15 | 17 | 60 |
| (7, 24, 25) | 7 | 24 | 25 | 84 |
| (9, 40, 41) | 9 | 40 | 41 | 180 |
| (20, 21, 29) | 20 | 21 | 29 | 210 |
Click any triple to auto-fill the calculator fields.
Examples
3-4-5 Triangle
a=3, b=4 → c = √(9+16) = √25 = 5. Area = 6, Perimeter = 12.
Find a Leg
Known leg = 5, Hypotenuse = 13 → a = √(169−25) = √144 = 12. This is the 5-12-13 triple.
Diagonal of a Rectangle
A rectangle 6m × 8m has diagonal = √(36+64) = √100 = 10m.
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². It was named after the Greek mathematician Pythagoras.
How do I find the hypotenuse?
Use the formula c = √(a² + b²). Square both legs, add them together, then take the square root. For example, if a=3 and b=4: c = √(9+16) = √25 = 5.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². The most well-known are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of these are also triples (e.g., 6-8-10).
Can I use the Pythagorean theorem for any triangle?
No. The Pythagorean theorem applies only to right triangles (triangles with a 90° angle). For non-right triangles, use the Law of Cosines instead.