Complete guide to the Pythagorean Theorem with step-by-step examples, the distance formula, and real-world applications in construction, navigation, and screen sizing.
What Is the Pythagorean Theorem?
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It's one of the most fundamental relationships in geometry, named after the ancient Greek mathematician Pythagoras.
How to Find the Hypotenuse
c = √(a² + b²). Example: a=3, b=4 → c = √(9 + 16) = √25 = 5. Real-world: A 40" TV has width 34.9" and height 19.6": diagonal = √(34.9² + 19.6²) = 40".
How to Find a Missing Leg
a = √(c² - b²). Example: c=13, b=5 → a = √(169 - 25) = √144 = 12. Real-world: A 20-ft ladder placed 5 ft from a wall reaches a height of √(400 - 25) = 19.4 ft.
The Distance Formula
Distance = √((x₂ - x₁)² + (y₂ - y₁)²). This is the Pythagorean theorem applied to coordinate geometry. The squared differences create a right triangle between any two points.
Real-World Applications
🏗️ Construction: The 3-4-5 rule ensures square corners. 📺 Screen Sizing: Diagonal measurement based on width and height. 🗺️ Navigation: Shortest distance between two points. 🚀 Physics: Vector magnitude and displacement calculations.
Pythagorean Triples
Common integer sets where a² + b² = c²: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41). These appear frequently in problems because they avoid irrational numbers.
Key Takeaways
- Formula: a² + b² = c² — only works for right triangles
- c is always the hypotenuse — the longest side opposite the right angle
- Real-world: Used in construction, navigation, screen sizing, and physics
- Memorize common triples like (3,4,5) and (5,12,13) for quick problem solving
Ready to calculate? Use our Pythagorean Theorem Calculator for instant solutions.