Pythagorean Theorem Calculator
This free Pythagorean theorem calculator solves the famous equation a² + b² = c² to find the hypotenuse or a missing leg of a right triangle. The Pythagorean theorem relates the three sides of any right-angled triangle, and it is one of the most useful results in all of geometry — used in construction, navigation, design, and math class. Enter any two sides and the calculator returns the third, with the steps shown.
How to Use the Pythagorean Theorem Calculator
- Enter the two sides you know — two legs, or one leg and the hypotenuse.
- Calculate — see the missing side and the working.
The Pythagorean Theorem
a² + b² = c²
Here a and b are the two shorter sides (legs) and c is the hypotenuse — the longest side, opposite the right angle. To find the hypotenuse: c = √(a² + b²). To find a missing leg: a = √(c² − b²).
Worked Examples
| Known Sides | Calculation | Result |
|---|---|---|
| a = 3, b = 4 | √(9 + 16) = √25 | c = 5 |
| a = 6, b = 8 | √(36 + 64) = √100 | c = 10 |
| c = 13, a = 5 | √(169 − 25) = √144 | b = 12 |
Common Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy the theorem. The best-known is 3-4-5, and its multiples (6-8-10, 9-12-15) also work. Other common triples include 5-12-13 and 8-15-17. Builders use the 3-4-5 method to check that corners are square.
Where the Pythagorean Theorem Is Used
- Construction: ensuring right angles and square corners.
- Navigation: finding straight-line (diagonal) distances.
- Geometry: the foundation for distance and trigonometry.
- Design and engineering: calculating lengths and clearances.
Frequently Asked Questions
What is the Pythagorean theorem?
It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
How do you find the hypotenuse?
Add the squares of the two legs and take the square root: c = √(a² + b²). For legs of 3 and 4, c = √25 = 5.
How do you find a missing leg?
Subtract the known leg's square from the hypotenuse's square and take the root: a = √(c² − b²). For c = 13 and b = 5, a = √144 = 12.
What is a Pythagorean triple?
It is a set of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, and 8-15-17.
Is this Pythagorean theorem calculator free?
Yes — it is completely free, requires no signup, and solves for any side with steps.