Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²).
Find the hypotenuse or either leg.
The Pythagorean theorem is one of the most fundamental relationships in mathematics, describing the exact relationship between the three sides of any right triangle.
The formula: a² + b² = c²
Where:
- a and b = the two legs of the right triangle (the sides adjacent to the 90° angle)
- c = the hypotenuse (the side opposite the right angle — always the longest side)
Solving for each variable:
- Hypotenuse: c = √(a² + b²)
- Unknown leg: a = √(c² − b²)
Worked examples:
Example 1 — Find the hypotenuse: A right triangle has legs of 3 and 4. c = √(3² + 4²) = √(9 + 16) = √25 = 5 This is the famous 3-4-5 Pythagorean triple.
Example 2 — Find a missing leg: The hypotenuse is 13 and one leg is 5. a = √(13² − 5²) = √(169 − 25) = √144 = 12
Common Pythagorean triples (whole-number solutions):
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Real-world applications:
- Construction: Checking that walls and corners are perfectly square (the 3-4-5 method)
- Navigation: Calculating straight-line distances from coordinates
- Screen size: A 16:9 screen with a 65-inch diagonal has a width of ~56.6 inches
- Surveying and mapping: Distance calculations on flat terrain
- Physics: Vector magnitude calculations (force, velocity components)
The theorem was known to Babylonian and Egyptian mathematicians thousands of years before Pythagoras, but he (or his school) is credited with the first general proof around 570–495 BC.