The Pythagorean theorem states that in a right triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the other two sides.
Example
A right triangle has legs of 3 and 4. Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5.
Applications
- Measuring diagonal distances
- Construction and carpentry (the 3-4-5 rule for square corners)
- Navigation and GPS calculations
- Computer graphics and game development
Common Pythagorean Triples
Certain whole-number side combinations satisfy a² + b² = c² exactly, which is why they show up constantly in construction and math problems: 3-4-5, 5-12-13, 8-15-17, and 7-24-25 (and any multiple of these, like 6-8-10). Builders use the 3-4-5 rule to check that a corner is exactly 90° without any angle-measuring tools.
The Converse Also Holds
If a² + b² = c² is true for a triangle's three sides, the triangle is guaranteed to have a right angle — this converse is what lets carpenters and surveyors verify square corners just by measuring the three sides, without a protractor or angle finder.