Pythagorean Triples for Builders: Beyond 3-4-5
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². Each one gives a right triangle with no fractions or decimals to measure — ideal when you are holding a tape in the wind. 3-4-5 is the famous one, but it is not always the best fit.
Useful triples
| Legs | Diagonal | Shape | When to use it |
|---|---|---|---|
| 3 – 4 | 5 | Near square | General layout |
| 5 – 12 | 13 | Long and narrow | Long walls, fence corners, narrow side yards |
| 8 – 15 | 17 | Long | Driveway edges, long slabs |
| 7 – 24 | 25 | Very long | Long building lines where space to the side is tight |
| 20 – 21 | 29 | Almost square | Square rooms and pads |
| 9 – 40 | 41 | Very long | Long property-line work |
| 12 – 35 | 37 | Long | Large slabs with limited side clearance |
Every multiple of a triple is also a triple: 10-24-26 is 5-12-13 doubled, 16-30-34 is 8-15-17 doubled.
Generating your own
For any two whole numbers m > n, the numbers m² − n², 2mn and m² + n² form a triple. With m = 2 and n = 1 you get 3, 4, 5. With m = 3 and n = 2 you get 5, 12, 13. With m = 4 and n = 1 you get 15, 8, 17.
Metric examples
In metric, scale by tenths: 0.5 / 1.2 / 1.3 m, 0.8 / 1.5 / 1.7 m, or 3 / 4 / 5 m. Because the triple is a ratio, the unit does not matter.
If none of the triples fits your space, use any two leg lengths and let the right triangle calculator give you the exact diagonal.
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Questions
Is 6-8-10 a different triple from 3-4-5?
It is the same triangle scaled by two. Builders call multiples by their own numbers because those are the numbers on the tape.