Pythagorean Theorem Calculator
Calculate any side of a right triangle using a² + b² = c²
Enter Values
Right Triangle Diagram
a² + b² = c²
where c is the hypotenuse
Common Pythagorean Triples
These are integer solutions where a² + b² = c² exactly.
Understanding the Pythagorean Theorem
The Pythagorean theorem is one of the most fundamental relationships in geometry. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
The Formula
a² + b² = c²
- a and b are the legs (the sides that form the right angle)
- c is the hypotenuse (the longest side, opposite the right angle)
Solving for Each Side
- Finding the hypotenuse: c = √(a² + b²)
- Finding a leg: a = √(c² - b²) or b = √(c² - a²)
The Converse
If a triangle has sides a, b, and c where a² + b² = c², then the triangle is a right triangle with the right angle opposite side c.
Applications
- Distance calculation in coordinate geometry
- Construction and architecture
- Navigation and surveying
- Computer graphics and game development
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a² + b² = c².
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle (90°). It's always the longest side of the right triangle and is typically labeled as 'c'.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). Multiples of triples are also triples.
Can I use this theorem for non-right triangles?
No, the Pythagorean theorem only applies to right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab·cos(C).
How do I prove a triangle is a right triangle?
If you know all three sides, check if a² + b² = c² (where c is the longest side). If true, it's a right triangle with the right angle opposite side c.
Why must c be greater than a and b?
The hypotenuse is always the longest side in a right triangle. If c ≤ a or c ≤ b, it's impossible to form a valid right triangle.
What's the 3D version of the Pythagorean theorem?
In 3D, the space diagonal d of a box with dimensions a, b, c is: d² = a² + b² + c², or d = √(a² + b² + c²).
How is this related to the distance formula?
The distance formula d = √[(x₂-x₁)² + (y₂-y₁)²] is a direct application of the Pythagorean theorem in the coordinate plane.