Pythagorean triples chart
A Pythagorean triple is three whole numbers that fit a² + b² = c², so they can be the sides of a right triangle: 3, 4 and 5 is the smallest. This chart lists the 16 primitive triples with c under 100.
Free printable PDF, 8th grade.
One page, black and white, prints the same every time.
What is on the sheet
| Triple (a, b, c) | a² + b² = c² | Times 2 | Times 3 |
|---|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6, 8, 10 | 9, 12, 15 |
| 5, 12, 13 | 25 + 144 = 169 | 10, 24, 26 | 15, 36, 39 |
| 8, 15, 17 | 64 + 225 = 289 | 16, 30, 34 | 24, 45, 51 |
| 7, 24, 25 | 49 + 576 = 625 | 14, 48, 50 | 21, 72, 75 |
| 20, 21, 29 | 400 + 441 = 841 | 40, 42, 58 | 60, 63, 87 |
| 12, 35, 37 | 144 + 1,225 = 1,369 | 24, 70, 74 | 36, 105, 111 |
| 9, 40, 41 | 81 + 1,600 = 1,681 | 18, 80, 82 | 27, 120, 123 |
| 28, 45, 53 | 784 + 2,025 = 2,809 | 56, 90, 106 | 84, 135, 159 |
| 11, 60, 61 | 121 + 3,600 = 3,721 | 22, 120, 122 | 33, 180, 183 |
| 16, 63, 65 | 256 + 3,969 = 4,225 | 32, 126, 130 | 48, 189, 195 |
| 33, 56, 65 | 1,089 + 3,136 = 4,225 | 66, 112, 130 | 99, 168, 195 |
| 48, 55, 73 | 2,304 + 3,025 = 5,329 | 96, 110, 146 | 144, 165, 219 |
| 13, 84, 85 | 169 + 7,056 = 7,225 | 26, 168, 170 | 39, 252, 255 |
| 36, 77, 85 | 1,296 + 5,929 = 7,225 | 72, 154, 170 | 108, 231, 255 |
| 39, 80, 89 | 1,521 + 6,400 = 7,921 | 78, 160, 178 | 117, 240, 267 |
| 65, 72, 97 | 4,225 + 5,184 = 9,409 | 130, 144, 194 | 195, 216, 291 |
Primitive triples and their multiples
A primitive triple has no factor shared by all three numbers. Every other triple is a primitive one multiplied by a whole number, so 6, 8, 10 and 9, 12, 15 come from 3, 4, 5. The last two columns show those first multiples; times 4, 5 and on up work the same way.
In every primitive triple one of the two shorter sides is even and the other is odd, and the hypotenuse is odd. One of the three is always a multiple of 5, which makes a quick check on a triple that looks wrong.
Using the chart with right triangles
Eighth grade uses the Pythagorean theorem to find a missing side (CCSS 8.G.B.7) and its converse to test whether a triangle is right (8.G.B.6). When two sides of a problem match a triple, or a multiple of one, the third side is on the chart and no square root is needed. A ladder 13 feet long with its foot 5 feet from the wall reaches 12 feet up.
Common questions
- Is 1, 1, 2 or 2, 3, 4 a Pythagorean triple?
- No. 1 + 1 is 2, not 4, and 4 + 9 is 13, not 16. Only numbers that fit a² + b² = c² exactly are triples, and the smallest is 3, 4, 5.
- How do you make a new Pythagorean triple?
- Pick two whole numbers m and n with m bigger than n. Then m² − n², 2mn and m² + n² is a triple. With m = 2 and n = 1 you get 3, 4, 5, and with m = 3 and n = 2 you get 5, 12, 13.