Pythagorean Triples Calculator

A Pythagorean triple is a set of three whole numbers that perfectly fit the rule a² + b² = c². When three numbers satisfy this equation, they can always form the sides of a right triangle. The most familiar example is 3, 4, 5 — but there are infinitely many others.

This calculator gives you two ways to work with Pythagorean triples. You can check whether any three numbers form a valid triple, or you can generate a new triple using the m and n formula. Both modes show full step-by-step solutions so you can follow the math.

Enter three side lengths to check a triple, or use m and n values to generate one. The calculator validates your inputs and shows detailed solution steps.

Formula

Every Pythagorean triple satisfies the equation a² + b² = c², where c is the hypotenuse and a and b are the two legs. If all three values are whole numbers and the equation holds true, you have a valid Pythagorean triple.

You can also generate triples using two positive integers m and n, where m is greater than n. The formulas a = m² − n², b = 2mn, and c = m² + n² will always produce a valid triple. When m and n share no common factors and are not both odd, the result is called a primitive triple.

How to Use This Calculator

1. Select a mode: Check a triple or Generate a triple.
2. For checking: enter three side values with c as the largest.
3. For generating: enter two positive integers m and n where m is greater than n.
4. Click Calculate to see the result and step-by-step solution.
5. Click Reset to clear all fields and start over.

Step-by-Step Examples

What the Result Means

When checking a triple, the calculator tells you whether the three numbers satisfy a² + b² = c². If the left side equals the right side, the numbers form a valid Pythagorean triple and can be the sides of a right triangle.

When generating a triple, the calculator produces three whole numbers using the m and n formulas. These numbers are guaranteed to form a right triangle. If m and n share no common factor and are not both odd, the triple is primitive — meaning it cannot be simplified further.

Side Ratio

A Pythagorean triple has whole-number side lengths in the form a : b : c.

The exact ratio changes from one triple to another, but every valid triple satisfies a² + b² = c² with c as the hypotenuse.

When to Use This Calculator

• Checking homework answers involving right triangle side lengths.
• Verifying whether measured side lengths form a true right angle.
• Generating triples for math competitions and practice problems.
• Building a reference list of Pythagorean triples for study.
• Confirming construction measurements that rely on the 3-4-5 rule or similar patterns.

Common Mistakes

• Placing c as a smaller side instead of the largest: c must always be the hypotenuse.
• Forgetting to square each side before adding: checking a + b = c instead of a² + b² = c².
• Writing the equation as a² + c² = b² instead of a² + b² = c².
• Using non-integer values and expecting a Pythagorean triple: triples must be whole numbers.
• Setting m less than or equal to n when generating: m must be strictly greater than n.