Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using a² + b² = c².
Understanding the Pythagorean Theorem
The Pythagorean theorem is one of the most fundamental and well-known theorems in mathematics. It describes the relationship between the sides of a right triangle and has been used for over 2,500 years in construction, navigation, and countless other applications.
The Formula
a² + b² = c²
Where a and b are the legs (shorter sides) and c is the hypotenuse (longest side, opposite the right angle)
Solving for Each Side
| Find | Formula | Example |
|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | a=3, b=4: c = √(9+16) = √25 = 5 |
| Side a | a = √(c² - b²) | b=4, c=5: a = √(25-16) = √9 = 3 |
| Side b | b = √(c² - a²) | a=3, c=5: b = √(25-9) = √16 = 4 |
Pythagorean Triples
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem. The most famous is (3, 4, 5).
Common Pythagorean Triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Scaled Triples (3-4-5 family)
| a | b | c | Factor |
|---|---|---|---|
| 6 | 8 | 10 | ×2 |
| 9 | 12 | 15 | ×3 |
| 12 | 16 | 20 | ×4 |
| 15 | 20 | 25 | ×5 |
Real-World Applications
- Construction: Ensuring walls are square (3-4-5 rule)
- Navigation: Calculating shortest distance
- Ladders: Finding safe ladder placement distances
- Television Sizing: Calculating diagonal screen size
- Sports Fields: Measuring diagonal distances
- Architecture: Designing roofs and ramps
Historical Context
While named after the Greek mathematician Pythagoras (c. 570-495 BCE), this relationship was known to ancient Babylonians and Egyptians long before. The theorem has hundreds of different proofs, more than any other mathematical theorem.
Visual Guide
Right triangle with legs a, b and hypotenuse c
Quick Check
To verify a right triangle, check if:
a² + b² = c²
Example: 3² + 4² = 9 + 16 = 25 = 5²