Pythagorean Triples Calculator

A Pythagorean triple consists of three positive integers a, b, and c, such that a squared plus b squared equals c squared (a^2 + b^2 = c^2). This relationship corresponds to the side lengths of a right-angled triangle. This calculator leverages Euclid's formula to generate primitive triples using two integers, m and n, where m > n > 0. The generated legs and hypotenuse are calculated as follows: a = m^2 - n^2, b = 2mn, and c = m^2 + n^2. Furthermore, scaling factors (k) can be applied to generate similar non-primitive triples, while verification modes check user-submitted values against the identity equation.

Let us generate a Pythagorean triple using Euclid's formula with the integers m = 4 and n = 2, with a scaling factor k = 1. First, calculate leg a by squaring m and n: a = (4^2 - 2^2) = 16 - 4 = 12. Next, calculate leg b by multiplying 2, m, and n: b = 2(4)(2) = 16. Finally, calculate the hypotenuse c by adding their squares: c = (4^2 + 2^2) = 16 + 4 = 20. This produces the classic triple (12, 16, 20), which is a scaled version of the 3-4-5 right triangle, satisfying 12^2 + 16^2 = 144 + 256 = 400, matching 20^2.

When inputting custom numbers to verify a triangle, ensure that the largest value is assigned as the hypotenuse (c) to prevent negative absolute differences. Always check if your generated triple is primitive by ensuring that the greatest common divisor (GCD) of your generator inputs m and n is 1, and that m and n have opposite parities (one is even, the other is odd).