The Pythagorean theorem is one of the most important relationships in mathematics, used to find the hypotenuse of a right triangle and solve countless real-world problems. Whether you're building, navigating, or solving geometry problems, understanding how to calculate the hypotenuse is essential.
The Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side opposite the right angle) equals the sum of the squares of the other two sides.
a² + b² = c²
Where:
a = first side (leg)
b = second side (leg)
c = hypotenuse (longest side)
Finding the Hypotenuse
To find the hypotenuse when you know both legs:
c = √(a² + b²)
Example 1: Right triangle with legs 3 and 4
c = √(3² + 4²)
c = √(9 + 16)
c = √25
c = 5
Example 2: Right triangle with legs 5 and 12
c = √(5² + 12²)
c = √(25 + 144)
c = √169
c = 13
Example 3: Right triangle with legs 6 and 8
c = √(6² + 8²)
c = √(36 + 64)
c = √100
c = 10
Common Pythagorean Triples
Pythagorean triples are sets of three whole numbers that satisfy the theorem. Memorizing these speeds up calculations:
| Side A | Side B | Hypotenuse | Multiple |
|---|---|---|---|
| 3 | 4 | 5 | 3-4-5 |
| 5 | 12 | 13 | 5-12-13 |
| 8 | 15 | 17 | 8-15-17 |
| 6 | 8 | 10 | Double 3-4-5 |
| 9 | 12 | 15 | Triple 3-4-5 |
| 7 | 24 | 25 | 7-24-25 |
| 20 | 21 | 29 | 20-21-29 |
| 9 | 40 | 41 | 9-40-41 |
Finding Missing Legs
If you know the hypotenuse and one leg, find the other:
a = √(c² - b²)
Example: Hypotenuse is 13, one leg is 5
a = √(13² - 5²)
a = √(169 - 25)
a = √144
a = 12
Practical Worked Examples
Example 1: Ladder Problem
A ladder leans against a wall 8 feet high.
The base is 6 feet from the wall.
What is the ladder length (hypotenuse)?
c = √(8² + 6²)
c = √(64 + 36)
c = √100
c = 10 feet
Example 2: Diagonal of a Rectangle
A rectangular field is 50 meters long and 30 meters wide.
What is the diagonal distance?
c = √(50² + 30²)
c = √(2500 + 900)
c = √3400
c ≈ 58.3 meters
Example 3: Construction Square
A building has a foundation 60 feet long and 40 feet wide.
To check if corners are square (90°), measure the diagonal.
Should be: c = √(60² + 40²) = √(3600 + 1600) = √5200 ≈ 72.1 feet
Real-World Applications
The Pythagorean theorem applies to:
- Construction: Checking right angles, finding roof rafter lengths
- Navigation: Calculating straight-line distances between points
- Sports: Determining distances across fields or courses
- Engineering: Stress calculations and structural design
- Surveying: Land measurement and mapping
- Technology: Screen diagonal measurements (16:9 aspect ratio)
Distance Formula in Coordinate Geometry
The Pythagorean theorem extends to finding distances between points:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Example: Distance between points (1, 2) and (4, 6)
Distance = √[(4-1)² + (6-2)²]
Distance = √[3² + 4²]
Distance = √[9 + 16]
Distance = √25
Distance = 5 units
The 3-4-5 Triangle Rule
The 3-4-5 right triangle is the most useful Pythagorean triple. Contractors often use this rule to ensure corners are square: measure 3 feet along one wall, 4 feet along the perpendicular wall, and the diagonal should be exactly 5 feet.
Beyond Right Triangles
For non-right triangles, use the Law of Cosines instead:
c² = a² + b² - 2ab·cos(C)
Where C is the angle between sides a and b.
Use our Pythagorean Theorem Calculator to instantly find hypotenuse lengths and verify right angles.