HesapHanem

Pythagorean Triple Generator

Enter integers m and n (m > n ≥ 1) to generate a Pythagorean triple.

How it is calculated

The tool applies Euclid's formula to two whole numbers m and n with m > n ≥ 1:

a = m² − n², b = 2mn, c = m² + n²

These always satisfy a² + b² = c², so they are the two legs and the hypotenuse of a right triangle with whole-number sides. Decimals are rejected, and so is any pair where m is not larger than n or n is below 1. The tool does not check that the triple is primitive. If m and n share a factor, or are both odd, the three numbers will have a common divisor (for m = 5 and n = 3 you get 16, 30, 34, which is 2 times 8, 15, 17).

Worked example

Choose m = 5 and n = 2. Then a = 25 − 4 = 21, b = 2 × 5 × 2 = 20 and c = 25 + 4 = 29. Check: 21² + 20² = 441 + 400 = 841, and 29² = 841. So a triangle with sides 20, 21 and 29 has a right angle. Because 5 and 2 are coprime and of opposite parity, this triple is primitive.

Try this example →

Common mistakes

  • Entering m smaller than or equal to n; the order matters, m must be the larger number.
  • Typing decimals like 2.5, which are rejected. Both inputs must be whole numbers.
  • Assuming every result is a primitive triple. Pairs such as 5 and 3 give 16, 30, 34, which reduces to 8, 15, 17.
  • Expecting every triple to appear from a single pair. Euclid's formula gives primitive ones; non-primitive ones come from multiplying all three numbers by k, for example 6, 8, 10 from 3, 4, 5.
  • Mixing up which number is the hypotenuse: c is always the largest, and a and b are the legs, though a may be larger or smaller than b.

About this calculator

This calculator generates Pythagorean triples using Euclid's formula: for any two positive integers m > n, the triple a = m² − n², b = 2mn, c = m² + n² always satisfies a² + b² = c². Choosing m and n coprime and not both odd yields every primitive Pythagorean triple (other choices, such as m = 3, n = 1 giving 8-6-10, give a multiple of a primitive triple), and scaling the result by any integer k produces all non-primitive ones too.

Pythagorean triples describe right triangles with whole-number side lengths — like the well-known 3-4-5 triangle — and are used in geometry education, construction layout (squaring corners without a protractor), and number theory.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    A practical use is squaring a corner on site. The pair m = 2, n = 1 gives the 3-4-5 triple, and m = 3, n = 2 gives 5-12-13. Multiply every side by the same number to work at full scale: 3-4-5 becomes 9-12-15 or 30-40-50, and the corner is square when your three measurements match. Remember the tool gives only one triple per pair. Which pair of numbers produced a triple you liked, and what are you laying out?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators