Pythagorean Theorem Calculator
Find any missing side of a right triangle using a² + b² = c². Solves for a, b or hypotenuse c.
Inputs
Missing side
5.00000000
Leg a
3.00000000
Leg b
4.00000000
Hypotenuse c
5.00000000
Verification
3² + 4² = 5² → 9 + 16 = 25
Step by step
Write the theorem
a² + b² = c²
Substitute
3² + 4² = c² → 9 + 16 = c²
Solve for Hypotenuse c
c = √(3² + 4²) = √(9 + 16) = √25
= 5
How it works
The Pythagorean theorem states that in any right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root. To find a missing leg, subtract the square of the known leg from the square of the hypotenuse and take the square root. Famous integer solutions (Pythagorean triples) include 3-4-5, 5-12-13, 8-15-17 and 7-24-25.
Formulas
Pythagorean theorem
a² + b² = c²
- a
- Leg a
- b
- Leg b
- c
- Hypotenuse (longest side)
Solve for hypotenuse
c = √(a² + b²)
Solve for a leg
a = √(c² − b²)
Frequently Asked Questions
What are Pythagorean triples?
A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c². The most famous is 3-4-5 (9 + 16 = 25). Others include 5-12-13, 8-15-17, and any multiple of these, like 6-8-10.
Does the theorem work in 3D?
Yes. The 3D diagonal d of a box with dimensions a, b, c is d = √(a² + b² + c²), which applies the Pythagorean theorem twice — once for the floor diagonal, then again for the space diagonal.
Why does the hypotenuse have to be the longest side?
The right angle is the largest angle in a right triangle. In any triangle, the longest side is opposite the largest angle. So the side opposite 90° (the hypotenuse) is always the longest.