Pythagorean Triples List (Primitive & Common)
A Pythagorean triple is a set of three whole numbers where a squared plus b squared equals c squared. This list shows the classic primitive triples, the multiples of the 3-4-5 family, and a live hypotenuse solver.
Pythagorean Theorema² + b² = c²c is the hypotenuse, opposite the right angle
Find the Hypotenusec = √(a² + b²)Needs: the two legs a and b
Find a Missing Lega = √(c² − b²)Needs: hypotenuse c and the other leg b
Euclid's Generator(m²−n², 2mn, m²+n²)Needs: whole numbers m > n > 0
Primitive Triplegcd(a, b, c) = 1No common factor, like (5, 12, 13)
Multiple of a Triple(ka, kb, kc)Scale any triple by a whole number k
Hypotenuse Solver & Triple Checker
Hypotenuse c = 5
Tip: open this page with ?c=25 to list every whole-number right triangle with hypotenuse 25 and highlight it in the tables. In Check a Triple the largest number is treated as the hypotenuse c.
| Triple (a, b, c) | Type | a² + b² | c² | Hypotenuse (c) | Copy |
|---|---|---|---|---|---|
| (3, 4, 5) | Primitive | 9 + 16 = 25 | 25 | 5 | |
| (5, 12, 13) | Primitive | 25 + 144 = 169 | 169 | 13 | |
| (6, 8, 10) | Multiple of (3,4,5) | 36 + 64 = 100 | 100 | 10 | |
| (7, 24, 25) | Primitive | 49 + 576 = 625 | 625 | 25 | |
| (8, 15, 17) | Primitive | 64 + 225 = 289 | 289 | 17 | |
| (9, 40, 41) | Primitive | 81 + 1600 = 1681 | 1681 | 41 | |
| (11, 60, 61) | Primitive | 121 + 3600 = 3721 | 3721 | 61 | |
| (12, 35, 37) | Primitive | 144 + 1225 = 1369 | 1369 | 37 |
| Triple (a, b, c) | Euclid (m, n) | a² + b² | c² | Hypotenuse (c) | Area (a × b / 2) | Copy |
|---|---|---|---|---|---|---|
| (3, 4, 5) | 2, 1 | 9 + 16 = 25 | 25 | 5 | 6 | |
| (5, 12, 13) | 3, 2 | 25 + 144 = 169 | 169 | 13 | 30 | |
| (8, 15, 17) | 4, 1 | 64 + 225 = 289 | 289 | 17 | 60 | |
| (7, 24, 25) | 4, 3 | 49 + 576 = 625 | 625 | 25 | 84 | |
| (20, 21, 29) | 5, 2 | 400 + 441 = 841 | 841 | 29 | 210 | |
| (12, 35, 37) | 6, 1 | 144 + 1225 = 1369 | 1369 | 37 | 210 | |
| (9, 40, 41) | 5, 4 | 81 + 1600 = 1681 | 1681 | 41 | 180 | |
| (28, 45, 53) | 7, 2 | 784 + 2025 = 2809 | 2809 | 53 | 630 | |
| (11, 60, 61) | 6, 5 | 121 + 3600 = 3721 | 3721 | 61 | 330 | |
| (16, 63, 65) | 8, 1 | 256 + 3969 = 4225 | 4225 | 65 | 504 | |
| (33, 56, 65) | 7, 4 | 1089 + 3136 = 4225 | 4225 | 65 | 924 | |
| (48, 55, 73) | 8, 3 | 2304 + 3025 = 5329 | 5329 | 73 | 1320 | |
| (13, 84, 85) | 7, 6 | 169 + 7056 = 7225 | 7225 | 85 | 546 | |
| (36, 77, 85) | 9, 2 | 1296 + 5929 = 7225 | 7225 | 85 | 1386 | |
| (39, 80, 89) | 8, 5 | 1521 + 6400 = 7921 | 7921 | 89 | 1560 | |
| (65, 72, 97) | 9, 4 | 4225 + 5184 = 9409 | 9409 | 97 | 2340 |
| Multiplier (k) | Triple (3k, 4k, 5k) | a² + b² | c² | Hypotenuse (c) | Copy |
|---|---|---|---|---|---|
| 1 | (3, 4, 5) | 9 + 16 = 25 | 25 | 5 | |
| 2 | (6, 8, 10) | 36 + 64 = 100 | 100 | 10 | |
| 3 | (9, 12, 15) | 81 + 144 = 225 | 225 | 15 | |
| 4 | (12, 16, 20) | 144 + 256 = 400 | 400 | 20 | |
| 5 | (15, 20, 25) | 225 + 400 = 625 | 625 | 25 | |
| 6 | (18, 24, 30) | 324 + 576 = 900 | 900 | 30 | |
| 7 | (21, 28, 35) | 441 + 784 = 1225 | 1225 | 35 | |
| 8 | (24, 32, 40) | 576 + 1024 = 1600 | 1600 | 40 | |
| 9 | (27, 36, 45) | 729 + 1296 = 2025 | 2025 | 45 | |
| 10 | (30, 40, 50) | 900 + 1600 = 2500 | 2500 | 50 | |
| 11 | (33, 44, 55) | 1089 + 1936 = 3025 | 3025 | 55 | |
| 12 | (36, 48, 60) | 1296 + 2304 = 3600 | 3600 | 60 | |
| 13 | (39, 52, 65) | 1521 + 2704 = 4225 | 4225 | 65 | |
| 14 | (42, 56, 70) | 1764 + 3136 = 4900 | 4900 | 70 | |
| 15 | (45, 60, 75) | 2025 + 3600 = 5625 | 5625 | 75 | |
| 16 | (48, 64, 80) | 2304 + 4096 = 6400 | 6400 | 80 | |
| 17 | (51, 68, 85) | 2601 + 4624 = 7225 | 7225 | 85 | |
| 18 | (54, 72, 90) | 2916 + 5184 = 8100 | 8100 | 90 | |
| 19 | (57, 76, 95) | 3249 + 5776 = 9025 | 9025 | 95 | |
| 20 | (60, 80, 100) | 3600 + 6400 = 10000 | 10000 | 100 |
Save & Share This List
Frequently Asked Questions
| Question | Answer |
|---|---|
| What is a Pythagorean triple? | A Pythagorean triple is a set of three positive whole numbers (a, b, c) with a² + b² = c². For (3, 4, 5): 9 + 16 = 25. |
| What is a primitive Pythagorean triple? | A primitive triple has no common factor above 1, so gcd(a, b, c) = 1. Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25). (6, 8, 10) is not primitive because it is 2 x (3, 4, 5). |
| How do you generate Pythagorean triples? | Use Euclid's formula: for whole numbers m > n > 0, the triple is (m² - n², 2mn, m² + n²). With m = 2 and n = 1 you get (3, 4, 5); with m = 3 and n = 2 you get (5, 12, 13). |
| What is the 3-4-5 triangle family? | It is every multiple k x (3, 4, 5), such as (6, 8, 10), (9, 12, 15), (12, 16, 20) and (15, 20, 25). All of them are right triangles with the same shape as the 3-4-5 triangle. |
| How many Pythagorean triples have a hypotenuse up to 100? | There are 16 primitive triples and 52 triples in total (primitive plus multiples) with hypotenuse c up to 100. |
| How do you find the hypotenuse of a right triangle? | Use c = sqrt(a² + b²). For legs 5 and 12: 25 + 144 = 169, and sqrt(169) = 13, so the triple is (5, 12, 13). |