Pythagorean Theorem Calculator
a² + b² = c² — find any side of a right triangle instantly
The Pythagorean Theorem: a² + b² = c²
In any right triangle, the square of the hypotenuse (the longest side, opposite the 90° angle) equals the sum of the squares of the two legs. This formula was known to ancient Babylonians, but is named after the Greek mathematician Pythagoras (c. 570–495 BC).
a = √(c² − b²) — find leg a
b = √(c² − a²) — find leg b
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers (a, b, c) where a² + b² = c². Click any row to load it.
| a | b | c (hyp.) | Notes |
|---|---|---|---|
| 3 | 4 | 5 | Most common triple |
| 6 | 8 | 10 | × 2 of 3-4-5 |
| 5 | 12 | 13 | Second most common |
| 8 | 15 | 17 | — |
| 7 | 24 | 25 | — |
| 9 | 12 | 15 | × 3 of 3-4-5 |
| 20 | 21 | 29 | — |
| 9 | 40 | 41 | — |
| 12 | 35 | 37 | — |
| 11 | 60 | 61 | — |
How to Use This Calculator
Select which side you want to find using the buttons at the top of the calculator:
- Find c — enter legs a and b, calculator finds hypotenuse c
- Find a — enter leg b and hypotenuse c, calculator finds leg a
- Find b — enter leg a and hypotenuse c, calculator finds leg b
Real-world use: The 3-4-5 triangle is used in construction to ensure corners are exactly 90°. Measure 3 feet along one wall, 4 feet along the other — if the diagonal is 5 feet, the corner is perfectly square.
Triangle measurements
The calculator also shows:
- Area = (a × b) / 2
- Perimeter = a + b + c
- Angle α = arctan(a / b) — the angle opposite leg a
- Angle β = 90° − α (since angles in a triangle sum to 180°)
Real-World Applications of the Pythagorean Theorem
- Construction & carpentry — checking square corners, calculating rafter lengths, staircase design, and diagonal bracing.
- Navigation — finding the straight-line distance between two points on a map using the distance formula (derived from the Pythagorean theorem).
- Screen & display dimensions — a TV's diagonal size is calculated from width and height using a² + b² = c².
- Physics & engineering — calculating resultant forces, velocity components, and vector magnitudes in 2D space.
- Computer graphics — calculating the distance between two points in a 2D or 3D coordinate system.
Frequently Asked Questions
What is the Pythagorean theorem?
In a right triangle with legs a and b and hypotenuse c: a² + b² = c². The hypotenuse is always the side opposite the 90° angle.
How do I find the hypotenuse?
Use: c = √(a² + b²). Example: a = 3, b = 4 → c = √(9 + 16) = √25 = 5.
How do I find a missing leg?
Use: a = √(c² − b²) or b = √(c² − a²). Example: c = 5, b = 4 → a = √(25 − 16) = √9 = 3.
What are Pythagorean triples?
Sets of three integers where a² + b² = c². The most common: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples work too: 6-8-10, 9-12-15.
Does the theorem work for all triangles?
No — only for right triangles (triangles with exactly one 90° angle). For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
What is the hypotenuse of a 45-45-90 triangle with legs of 1?
c = √(1² + 1²) = √2 ≈ 1.414. In a 45-45-90 triangle, the hypotenuse is always leg × √2.