The Pythagorean theorem is one of the most famous and useful formulas in mathematics. Named after the ancient Greek mathematician Pythagoras, it describes the relationship between the sides of a right triangle.
The Formula
a² + b² = c²
Where:
- a and b are the two legs (shorter sides)
- c is the hypotenuse (longest side, opposite the right angle)
How to Use the Pythagorean Theorem
1. Find the Hypotenuse (given a and b)
c = √(a² + b²)
*Example: a=3, b=4 → c = √(9+16) = √25 = 5*
2. Find a Leg (given one leg and the hypotenuse)
a = √(c² - b²)
*Example: c=10, b=6 → a = √(100-36) = √64 = 8*
Real-World Applications
The Pythagorean theorem isn't just for math class — it's used every day:
📐 Construction: Ensuring corners are square. A 3-4-5 triangle verifies a 90° angle.
🏠 Home Improvement: Calculating the length of a diagonal brace, the height of a roof, or the size of a TV screen.
🗺 Navigation: Finding the shortest distance between two points on a map.
🎮 Game Development: Calculating distances in 2D and 3D games.
📡 Telecommunications: Antenna height calculations for optimal signal range.
Common Pythagorean Triples
These sets of whole numbers satisfy the theorem:
- 3-4-5: 3² + 4² = 5² (9+16=25)
- 5-12-13: 5² + 12² = 13² (25+144=169)
- 8-15-17: 8² + 15² = 17² (64+225=289)
- 7-24-25: 7² + 24² = 25² (49+576=625)
Try Our Pythagorean Theorem Calculator
Our Pythagorean Theorem Calculator lets you:
- Enter any two sides of a right triangle
- Find the third side instantly
- Also calculates area and perimeter
- Perfect for students, builders, and DIY enthusiasts
💡 Pro Tip: In construction, the 3-4-5 rule is a quick way to check if a corner is perfectly square. Measure 3 feet along one wall, 4 feet along the other — the diagonal should be exactly 5 feet.
*Calculate missing triangle sides with our free Pythagorean Theorem Calculator.*
