Pythagorean Theorem
Definition
A fundamental theorem in geometry stating that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
Why is Pythagorean Theorem Important?
Pythagorean Theorem is a foundational mathematical concept used across science, engineering, finance, and everyday problem-solving. From analyzing data sets to optimizing business decisions, this concept provides the analytical framework needed to interpret quantitative information accurately.
Our math calculators make complex computations simple and accessible, providing step-by-step results that help students, professionals, and curious minds explore mathematical relationships with confidence.
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in any right 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
This is arguably the most well-known and widely applied theorem in all of mathematics, fundamental to geometry, trigonometry, physics, engineering, and navigation.
Solving for Each Side
| Find | Formula | Example (a=3, b=4) |
|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | c = √(9+16) = √25 = 5 |
| Leg (a) | a = √(c² − b²) | a = √(25−16) = √9 = 3 |
| Leg (b) | b = √(c² − a²) | b = √(25−9) = √16 = 4 |
Common Pythagorean Triples
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 12-16-20 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
Real-World Applications
- Construction: The 3-4-5 rule checks if corners are square (90°)
- Navigation: Distance between two GPS coordinates
- Screen size: 16" wide × 9" tall = √(256+81) = √337 ≈ 18.4" diagonal
- Ladder safety: 12' ladder against a wall, base 3' out → reaches √(144−9) = √135 ≈ 11.6' high