Pythagorean Theorem
Calculate the hypotenuse or any side of a right triangle using a² + b² = c².
The Pythagorean theorem explained
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. Discovered by ancient Greek mathematicians and named after Pythagoras (c. 570–495 BC), this relationship is one of the most important in all of mathematics.
a² + b² = c²
Where:
a, b = the two legs (sides adjacent to the right angle)
c = the hypotenuse (longest side, opposite the right angle)
Find hypotenuse: c = √(a² + b²)
Find leg a: a = √(c² − b²)
Find leg b: b = √(c² − a²)Pythagorean triples: whole-number right triangles
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². The 3-4-5 triangle is the simplest and most famous. Any multiple of a Pythagorean triple is also a valid triple — e.g. (6, 8, 10) and (9, 12, 15) are multiples of (3, 4, 5).
Real-world applications
The 3D extension: distance in space
The theorem extends to three dimensions. The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) in 3D space is found by applying the theorem twice. This is used in 3D graphics, physics simulations, and GPS altitude calculations.
2D distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
3D distance: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Example — diagonal of a box (3m × 4m × 12m):
Flat diagonal = √(3² + 4²) = √25 = 5m
Space diagonal = √(5² + 12²) = √169 = 13mFrequently asked questions
What is the Pythagorean theorem?
For a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
How do I find the hypotenuse?
Square both shorter sides, add them, and take the square root. For sides 3 and 4: √(3² + 4²) = √25 = 5.
Can I find a missing leg instead of the hypotenuse?
Yes. Rearrange the formula: a = √(c² − b²). The tool solves for whichever side you leave blank.
Does it only work for right triangles?
Yes. The Pythagorean theorem applies only to triangles with a 90-degree angle. For other triangles you need the law of cosines.
a² + b² = c² | Find any side: c=√(a²+b²), a=√(c²−b²), b=√(c²−a²)