Pythagorean Triples Calculator: Find Integer Right Triangles (2026)
Find and generate Pythagorean triples. Enter a side to find all integer right triangles containing it, or generate triples from m,n parameters.
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Put these formulas into practice with our instant, step-by-step Pythagorean Triples Calculator.
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem: a² + b² = c². These integer solutions to the most famous equation in mathematics have fascinated mathematicians for millennia and appear in applications from construction to computer science. Our free Pythagorean triples calculator helps you find and generate these triples instantly.
Key Takeaway
Pythagorean triples are integer solutions to a² + b² = c². The generator formula a = m²-n², b = 2mn, c = m²+n² produces all primitive triples for m > n > 0.
What are Pythagorean Triples?
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c². The most well-known example is (3, 4, 5), where 9 + 16 = 25. These triples correspond to right triangles where all three sides have integer lengths.
The Generator Formula
b = 2mn
c = m² + n²
where m > n > 0
This formula, attributed to Euclid, generates all primitive Pythagorean triples when m and n are coprime and not both odd.
Common Pythagorean Triples
| Triple | m, n | Verification |
|---|---|---|
| (3, 4, 5) | 2, 1 | 9 + 16 = 25 — |
| (5, 12, 13) | 3, 2 | 25 + 144 = 169 — |
| (8, 15, 17) | 4, 1 | 64 + 225 = 289 — |
| (7, 24, 25) | 4, 3 | 49 + 576 = 625 — |
Real-World Applications
- Construction: The 3-4-5 rule for verifying right angles
- Surveying: Measuring distances and angles in the field
- Computer Science: Distance calculations in games and graphics
- Education: Teaching the Pythagorean theorem
- Architecture: Ensuring square corners in building layouts
Frequently Asked Questions
What is a primitive Pythagorean triple?
A primitive triple has no common factors other than 1 (gcd(a,b,c) = 1). For example, (3, 4, 5) is primitive, but (6, 8, 10) is not (it is 2 × (3, 4, 5)).
How do I verify if three numbers form a Pythagorean triple?
Check if the largest number squared equals the sum of the squares of the other two. For (a, b, c) with c being the largest, verify a² + b² = c².
Try our Pythagorean triples calculator to find and generate triples instantly, or explore our right triangle calculator and triangle perimeter calculator for more triangle computations.