Understanding Pythagorean Theorem Calculator
The Pythagorean theorem states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
a² + b² = c²
Hypotenuse: c = √(a² + b²)
Leg a: a = √(c² − b²)
Leg b: b = √(c² − a²)
Pythagorean Triples — Perfect Integer Solutions
A Pythagorean triple consists of three positive integers a, b, c satisfying a²+b²=c². These are elegant solutions where all sides are whole numbers.
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 9+16=25 ✓ |
| 5 | 12 | 13 | 25+144=169 ✓ |
| 8 | 15 | 17 | 64+225=289 ✓ |
| 7 | 24 | 25 | 49+576=625 ✓ |
| 9 | 40 | 41 | 81+1600=1681 ✓ |
| 6 | 8 | 10 | 36+64=100 ✓ (3-4-5 × 2) |
| 20 | 21 | 29 | 400+441=841 ✓ |
Converse of the Pythagorean Theorem
The converse is equally powerful: if a² + b² = c², the triangle MUST be a right triangle. This is used to verify right angles in construction and carpentry.
Example — The 3-4-5 Rule: Builders use this to check if corners are perfectly square. Measure 3 units along one wall, 4 units along the adjacent wall. If the diagonal is exactly 5 units, the corner is a true right angle (90°).
Real-World Applications
- Construction & carpentry: Checking square corners, calculating rafter lengths, diagonal measurements
- Navigation: Finding straight-line distance between two points when you know north-south and east-west distances
- Screen sizes: A TV described as "55 inch" means its diagonal = 55 inches. If it's 16:9 aspect ratio, width = 47.95 in, height = 26.97 in, and 47.95²+26.97² = 55² ✓
- Physics: Vector addition, velocity components, force diagrams
- Coordinate geometry: The distance formula d = √((x₂-x₁)²+(y₂-y₁)²) is derived from the Pythagorean theorem