A Pythagorean triple is a set of three whole numbers that fit the Pythagorean theorem exactly — no rounding, no decimals. This calculator generates them from Euclid's formula, checks whether numbers you already have form one, and lists every triple that contains a given side, using the same relationship that has anchored right-triangle geometry since antiquity.

Why (3, 4, 5) is special

(3, 4, 5) is the smallest Pythagorean triple and the only one made of three consecutive integers. Because 3² + 4² = 9 + 16 = 25 = 5², a triangle with those side lengths always has a perfect 90° angle — which is why builders have used 3-4-5 (and its multiples, like 6-8-10) ropes and tape measures to square corners for thousands of years, long before anyone wrote down the algebra behind it.

Every multiple of (3, 4, 5) — (6, 8, 10), (9, 12, 15), (12, 16, 20), and so on forever — is also a valid triple, because scaling every side of a right triangle by the same factor preserves all its angles.

Euclid's formula generates them all

Euclid proved that every primitive Pythagorean triple can be generated from a pair of integers m and n (with m > n > 0, coprime, and opposite parity) via a = m² − n², b = 2mn, c = m² + n². Plugging in m = 2, n = 1 gives (3, 4, 5); m = 3, n = 2 gives (5, 12, 13); m = 4, n = 1 gives (8, 15, 17). Every non-primitive triple is then just a whole-number multiple of one of these.

This is more than a curiosity — it means the infinite set of Pythagorean triples has a complete, finite description: pick any valid (m, n) pair for the primitive triples, then scale by any positive integer for the rest.

Primitive vs. non-primitive, and how to tell them apart

A triple is primitive when gcd(a, b, c) = 1 — the three numbers share no common factor. (5, 12, 13) is primitive; (10, 24, 26) is not, because dividing every term by 2 gives back (5, 12, 13). To check whether a triple you found is primitive, find the greatest common divisor of all three numbers: if it's 1, you have a primitive triple; if it's k > 1, dividing by k reveals the primitive triple underneath.

This distinction matters when listing every triple containing a given side: the complete list always splits into a handful of primitive triples plus their scaled multiples, and this calculator's List by Side tab labels each result accordingly.