Pythagorean Theorem Calculator
Find any missing side of a right triangle, with the area, perimeter and full steps.
Results
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Steps
| Step | Working |
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Quick Answer
The Pythagorean theorem says a² + b² = c² in a right triangle, where c is the longest side. With legs of 3 and 4, c² = 9 + 16 = 25, so the hypotenuse is 5. To find a leg, subtract: with c = 13 and a = 5, b = √(169 − 25) = 12.
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Table of Contents
How This Pythagorean Theorem Calculator Works
In a right triangle, the two shorter sides are the legs, and the longest side is the hypotenuse. Enter any two sides, and the calculator finds the third. It also gives the area and perimeter.
Quick explanation: a² + b² = c².
- Legs a and b: the sides that form the right angle.
- Hypotenuse c: the side opposite the right angle.
- For other triangles: use our Triangle Calculator.
Core Formulas
- Hypotenuse: c = √(a² + b²)
- Missing leg: a = √(c² − b²)
- Area: a × b ÷ 2
- Perimeter: a + b + c
Worked Example
A ladder is 13 ft long, and its base is 5 ft from the wall. How high does it reach?
- b² = 13² − 5² = 169 − 25 = 144
- b = √144 = 12 ft
Common Pythagorean Triples
These whole-number sets are right triangles, so they make quick checks:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Frequently Asked Questions
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c².
Square both legs, add them, and take the square root.
Square the hypotenuse, subtract the square of the known leg, and take the square root.
No. It only works on right triangles. For others, use the Law of Cosines in our Triangle Calculator.
It is a set of three whole numbers that fit a² + b² = c², such as 3, 4 and 5.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Pythagorean theorem | The theorem and its use. | https://en.wikipedia.org/wiki/Pythagorean_theorem |
| Wikipedia, Pythagorean triple | Whole-number examples. | https://en.wikipedia.org/wiki/Pythagorean_triple |