Pythagorean Triples Calculator

What is a Pythagorean Triple?

A Pythagorean triple consists of three positive integers $(a, b, c)$ such that:

By the Pythagorean theorem, these three integers represent the exact side lengths of a right-angled triangle, where $a$ and $b$ are the perpendicular legs and $c$ is the hypotenuse. The most well-known Pythagorean triple is $(3, 4, 5)$, since $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. If you need to calculate floating-point side lengths or angles for any right triangle, explore our pythagorean calculator.

Primitive vs. Non-Primitive Triples

A Pythagorean triple $(a, b, c)$ is classified as a Primitive Pythagorean Triple (PPT) if $a$, $b$, and $c$ are mutually coprime, meaning their greatest common divisor is 1:

Any multiple of a primitive triple is also a valid Pythagorean triple, but not primitive. For instance, multiplying $(3, 4, 5)$ by 2 gives $(6, 8, 10)$, since $6^2 + 8^2 = 36 + 64 = 100 = 10^2$. Here $\gcd(6, 8, 10) = 2$, so $(6, 8, 10)$ is non-primitive.

Euclid's Formula for Generating Triples

Euclid discovered a universal formula in his Elements (Book X) to generate all primitive Pythagorean triples using two positive integers $m$ and $n$ with $m > n$:

• Leg 1: $a = m^2 - n^2$
• Leg 2: $b = 2mn$
• Hypotenuse: $c = m^2 + n^2$

The triple $(a, b, c)$ generated by Euclid's formula is primitive if and only if $m$ and $n$ are coprime ($\gcd(m, n) = 1$) and have opposite parity (one is odd, the other is even).

Table of Common Primitive Pythagorean Triples