Pythagorean triples – like 3-4-5, 5-12-13 and 8-15-17 – are whole-number sides of right triangles. Babylonian tablets list them, builders use them to set out right angles, and Euclid’s formula generates every primitive triple.
How to use the Pythagorean Triples Calculator
- Mode
- First number
- Second number
- Third number
- m
- n
- Multiplier k
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
In check mode the numbers are sorted and the smallest two squares are compared with the largest. In generate mode, m and n give a primitive triple when they are coprime and of opposite parity; k scales it.
Formula
a² + b² = c²
Euclid: a = k(m² − n²), b = 2kmn, c = k(m² + n²), m > n > 0
Worked example
Inputs — Mode: Generate a triple (Euclid's formula); m: 2; n: 1; Multiplier k: 1.
Results — Result = (3, 4, 5); Check = 3² + 4² = 25 = 5²; Primitive? = Primitive (no common factor).
Key terms
- Pythagorean triple
- Positive integers a, b, c with a² + b² = c².
Frequently asked questions
What is a primitive triple?
One with no common factor, like 3-4-5 (6-8-10 is not primitive).
Are there infinitely many triples?
Yes – Euclid's formula gives infinitely many primitive ones.
What is the smallest triple?
3, 4, 5.
Is one leg always even?
Yes, in every primitive triple exactly one leg is even.