The Pythagorean theorem

For any right triangle with legs a and b and hypotenuse c:

a² + b² = c²

How to use this calculator

Find hypotenuse (c): enter both legs a and b.
Find leg (a): enter leg b and hypotenuse c.
Find leg (b): enter leg a and hypotenuse c.
Check triple: enter all three sides to verify whether they satisfy the theorem.

Common Pythagorean triples

3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
20, 21, 29

Practical applications

The Pythagorean theorem is used in navigation, construction, physics, and computer graphics to calculate distances and verify right angles.

Generating Pythagorean triples

Euclid’s formula generates all primitive Pythagorean triples: for any integers m > n > 0 with m and n coprime and not both odd:

a = m² − n²,  b = 2mn,  c = m² + n²

m = 2, n = 1 → (3, 4, 5)
m = 3, n = 2 → (5, 12, 13)
m = 4, n = 1 → (15, 8, 17)
m = 4, n = 3 → (7, 24, 25)

3D distance extension

The Pythagorean theorem extends naturally to three dimensions:

d = √(a² + b² + c²)

This formula calculates the straight-line distance between two points in 3D space and is used in computer graphics (vertex distances), GPS (3D positioning), and robotics (end-effector reach). For points (x₁, y₁, z₁) and (x₂, y₂, z₂): d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²).

Construction applications

Builders use the 3-4-5 rule to verify right angles without a protractor: measure 3 ft along one wall from the corner, 4 ft along the adjacent wall, and if the diagonal is exactly 5 ft the corner is square. This technique is used for squaring building foundations, fence corners, and tile layouts. For large layouts, scale up: 6-8-10, 9-12-15, or any multiple of 3-4-5.