Pythagorean Triples Formula
Pythagorean triples formula comprises three integers that follow the rules defined by the Pythagoras theorem. The group of these triples is referred to as the Pythagorean triples and is commonly written in the form of (a, b, c). The triangle formed with the sides having these triples as the dimensions is called a Pythagorean triangle. Let us understand the Pythagorean triples formula in detail using solved examples in the following section.
What is Pythagorean Triples Formula
Pythagorean triples formula is used to find the triples or group of three terms that satisfy the Pythagoras theorem. We know that Pythagoras' theorem is given as;
c^{2} = a^{2}+b^{2}
where,
 a, b = Base, and perpendicular of a rightangled triangle
 c = Hypotenuse of a rightangled triangle
If "a and "b" are two sides of a triangle and "c" is the hypotenuse of the triangle, the Pythagorean triples formula can be given as,
 a = m^{2}n^{2}
 b = 2mn
 c = m^{2}+n^{2}
where,
 a, b = Base, and perpendicular of a rightangled triangle
 c = Hypotenuse of a rightangled triangle
 m, n and k are any two positive integers; m > n

m and n are coprime and both should not be odd numbers
Note: If any number of a Pythagorean triple is given, then the other two numbers can be generated by using, a = m^{2}−n^{2}, b = 2mn, and c = m^{2}+n^{2}.
Let us have a look at a few solved examples on the Pythagorean triples formula to understand the concept better.
Solved Examples Using Pythagorean Triples Formula

Example 1: Check if (5, 12, 13) is a Pythagorean triple.
Solution:
To find: Check whether (5, 12, 13) is a Pythagorean triple.
Given: a = 5; b = 12; c = 13
Using the Pythagorean triples formula, we know that a Pythagorean triple satisfies Pythagoras' theorem: c^{2} = a^{2}+b^{2}
L. H. S. = c^{2} = 13^{2} = 169
R. H. S. = a^{2 }+ b^{2 }= 5^{2 }+ 12^{2} = 25 + 144 = 169
Since the given values satisfy the Pythagoras' theorem, they are a Pythagorean triple.
Answer: (5, 12, 13) is a Pythagorean triple.

Example 2: Two sides of a rightangled triangle measure 12 units and 16 units, find the length of the hypotenuse.
Solution:
To find: Length of hypotenuse of the rightangled triangle
We know that the sides of a rightangled triangle are Pythagorean triples. Using the Pythagorean triples formula,
Hypotenuse = √(Base)^{2} + (Perpendicular)^{2}
Hypotenuse = √12^{2} + 16^{2} = 20 units
Answer: Length of the hypotenuse of the rightangled triangle = 20 units.