The Pythagorean Theorem: a² + b² = c²
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. It is one of the most applied formulas in mathematics — used in construction, navigation, physics, and computer graphics.
The Formula
c² = a² + b² (finding hypotenuse)
a² = c² − b² (finding a leg)
b² = c² − a² (finding a leg)Worked Examples
- a = 3, b = 4 → c = √(9+16) = √25 = 5 (the classic 3-4-5 triple)
- a = 5, b = 12 → c = √(25+144) = √169 = 13
- c = 10, a = 6 → b = √(100−36) = √64 = 8
Common Pythagorean Triples
- 3, 4, 5 and multiples: 6-8-10, 9-12-15
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
The Converse Test
If a² + b² = c², the triangle is right-angled. If a² + b² > c², it is acute. If a² + b² < c², it is obtuse. This is used in construction to verify square corners.
Real-World Use
Builders use the 3-4-5 method to ensure walls meet at 90°. GPS systems use the 3D extension (a²+b²+c²=d²) to calculate distances. Screen diagonals are calculated with Pythagoras from width and height.
Calculate triangle sides: Free Pythagorean Theorem Calculator