Pythagorean Theorem Calculator
Pythagorean Theorem Calculator (a² + b² = c²)
About Pythagorean Theorem Calculator
The Pythagorean Theorem
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², this is one of the most fundamental relationships in mathematics. It was known to ancient Babylonians over 3,000 years ago and formally proven by the Greek mathematician Pythagoras around 500 BC. Today it remains essential in construction, navigation, physics, engineering, computer graphics, and architecture.
How It Works
Given a right triangle with legs a and b and hypotenuse c: to find the hypotenuse, compute c = sqrt(a² + b²). To find a missing leg, compute a = sqrt(c² - b²). The theorem only applies to right triangles (those containing a 90-degree angle). Common Pythagorean triples include (3,4,5), (5,12,13), (8,15,17), and (7,24,25) where all three sides are integers.
Practical Applications
Builders use the 3-4-5 rule to verify right angles. GPS systems use the theorem for distance calculations. Architects determine diagonal measurements. Carpenters calculate rafter lengths. Screen sizes are measured diagonally using this theorem. Even video game programmers use it to calculate distances between two points in a coordinate system.
How to Use This Calculator
Enter any two sides of a right triangle and select which side you want to solve for. The calculator computes the missing side and shows the step-by-step solution. It also displays whether the triangle is a Pythagorean triple and verifies the result.
Frequently Asked Questions
Does the Pythagorean Theorem work for all triangles?
No, it only applies to right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab*cos(C), which generalizes the Pythagorean theorem.
What is a Pythagorean triple?
A set of three positive integers (a, b, c) that satisfy a² + b² = c². Examples: (3,4,5), (5,12,13), (8,15,17). Any multiple of a triple is also a triple.
How do I know which side is the hypotenuse?
The hypotenuse is always the longest side and always opposite the right angle (90-degree angle). If you know two sides, the hypotenuse is the largest value.
