Pythagorean Theorem Calculator · Practical guide
Common Pythagorean Triples List
Whole-number side sets like 3-4-5 and 5-12-13 satisfy a² + b² = c² exactly — no rounding, no decimals.
At a glance
Quick answer
A Pythagorean triple is a set of three whole numbers satisfying a² + b² = c² exactly. The most useful ones to memorize are 3-4-5, 5-12-13, 8-15-17, and 7-24-25 — recognize one in a problem and you can skip the square root entirely.
Interactive tool
Pythagorean Theorem Calculator
Enter your own values, then compare the result with the examples below.
Results
Formula: a² + b² = c²
The Core Triples
| a | b | c | Check: a² + b² = c² |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 9 | 40 | 41 | 81 + 1,600 = 1,681 |
| 12 | 35 | 37 | 144 + 1,225 = 1,369 |
| 11 | 60 | 61 | 121 + 3,600 = 3,721 |
Primitive vs. Scaled Triples
A primitive triple has no common factor among its sides — 3-4-5 is primitive, but 6-8-10 is just 3-4-5 doubled. Multiplying every side by the same number always produces another valid triple, so one memorized triple unlocks infinitely many: 9-12-15, 12-16-20, 15-20-25, and so on. This is why the 3-4-5 layout rule scales up cleanly for decks and foundations.
Using Triples to Work Faster
Spotting a triple turns a two-step problem into a lookup. Legs of 24 and 7? That is the 7-24-25 set, so the hypotenuse is 25 with no calculation. Hypotenuse 26 and leg 10? Recognize 2 × (5-12-13), so the missing leg is 24. Enter any three sides above and the calculator confirms whether the set is a valid right triangle — or flags it as acute or obtuse if a² + b² does not equal c².
Frequently Asked Questions
How do I know if three numbers are a Pythagorean triple?
Square the two smaller numbers and add them; if the sum equals the square of the largest, it is a triple. Enter all three sides in the calculator to check instantly.
Are there infinitely many Pythagorean triples?
Yes. Every multiple of a known triple works, and formulas like a = m² − n², b = 2mn, c = m² + n² generate new primitive triples for any integers m > n.
Why do triples matter outside of math class?
Builders use 3-4-5 to square corners, and programmers use triples for exact integer distances in games and graphics where rounding errors are unacceptable.
Is 1-1-√2 a Pythagorean triple?
No. It is a valid right triangle (the 45-45-90 triangle), but triples require all three sides to be whole numbers, and √2 is irrational.