# Heronian Triangle Generator

A Heronian triangle is one whose side lengths *and* area are integers. They are named after the Greek mathematician Heron of Alexandria, whose name is also attached to the formula for triangular area in terms of side lengths (Heron's Formula).

Pythagorean triangles (right triangles whose sides are all integers) are the most prominent example of Heronian triangles, but they are not the only kind. For example, if you take two Pythagorean triangles that have a common leg length and put them together, you will create a new triangle that has integers side lengths and area. In fact, every Heronian triangle can be broken into two right triangles whose sides and areas are rational numbers.

Just as you can generate Pythagorean triangles with a two-parameter generator system, you can also find Heronian triangles with a three-parameter generator set:

Let m, n, and k be positive integers, and let

x = n(m² + k²)

y = m(n² + k²)

z = (m + n)|mn - k²|

Then x, y, and z form the sides of a Heronian triangle whose area is given by the formula

Area = mnk(m + n)|mn - k²|

*multiple*of every primitive Heronian triangle. For example, if you choose m = 2, n = 3, and k = 2, you generate the triangle with sides 10, 24, and 26.

The values of m and n can be switched with no effect on the generated triangle, but if you exchange m and k, or n and k, a different Heronian triangle will be generated. For instance, choosing m = 2, n = 2, and k = 3 yields a Heronian triangle with sides 10, 26, and 26.

Here is a table of primitive Heronian triangles ordered by increasing area. See also Nearly Equilateral Heronian Triangles.

Shortest Side |
Longest Side |
Perimeter | Area | |

3 | 4 | 5 | 12 | 6 |

5 | 5 | 6 | 16 | 12 |

5 | 5 | 8 | 18 | 12 |

4 | 13 | 15 | 32 | 24 |

5 | 12 | 13 | 30 | 30 |

9 | 10 | 17 | 36 | 36 |

3 | 25 | 26 | 54 | 36 |

7 | 15 | 20 | 42 | 42 |

10 | 13 | 13 | 36 | 60 |

8 | 15 | 17 | 40 | 60 |

13 | 13 | 24 | 50 | 60 |

6 | 25 | 29 | 60 | 60 |

11 | 13 | 20 | 44 | 66 |

5 | 29 | 30 | 64 | 72 |

13 | 14 | 15 | 42 | 84 |

10 | 17 | 21 | 48 | 84 |

7 | 24 | 25 | 56 | 84 |

8 | 29 | 35 | 72 | 84 |

12 | 17 | 25 | 54 | 90 |

4 | 51 | 53 | 108 | 90 |

19 | 20 | 37 | 76 | 114 |

16 | 17 | 17 | 50 | 120 |

17 | 17 | 30 | 64 | 120 |

16 | 25 | 39 | 80 | 120 |

13 | 20 | 21 | 54 | 126 |

15 | 28 | 41 | 84 | 126 |

5 | 51 | 52 | 108 | 126 |

11 | 25 | 30 | 66 | 132 |

15 | 26 | 37 | 78 | 156 |

13 | 40 | 51 | 104 | 156 |

14 | 25 | 25 | 64 | 168 |

10 | 35 | 39 | 84 | 168 |

25 | 25 | 48 | 98 | 168 |

13 | 30 | 37 | 80 | 180 |

9 | 40 | 41 | 90 | 180 |

12 | 55 | 65 | 132 | 198 |

17 | 25 | 26 | 68 | 204 |

20 | 21 | 29 | 70 | 210 |

17 | 25 | 28 | 70 | 210 |

17 | 28 | 39 | 84 | 210 |

12 | 35 | 37 | 84 | 210 |

7 | 65 | 68 | 140 | 210 |

3 | 148 | 149 | 300 | 210 |

9 | 73 | 80 | 162 | 216 |

15 | 41 | 52 | 108 | 234 |

13 | 37 | 40 | 90 | 240 |

15 | 34 | 35 | 84 | 252 |

13 | 40 | 45 | 98 | 252 |

9 | 65 | 70 | 144 | 252 |

15 | 37 | 44 | 96 | 264 |

33 | 34 | 65 | 132 | 264 |

27 | 29 | 52 | 108 | 270 |

17 | 65 | 80 | 162 | 288 |

25 | 51 | 74 | 150 | 300 |

5 | 122 | 123 | 250 | 300 |

20 | 37 | 51 | 108 | 306 |

17 | 39 | 44 | 100 | 330 |

25 | 33 | 52 | 110 | 330 |

11 | 60 | 61 | 132 | 330 |

11 | 100 | 109 | 220 | 330 |

17 | 40 | 41 | 98 | 336 |

24 | 35 | 53 | 112 | 336 |

15 | 52 | 61 | 128 | 336 |

4 | 193 | 195 | 392 | 336 |

25 | 29 | 36 | 90 | 360 |

18 | 41 | 41 | 100 | 360 |

41 | 41 | 80 | 162 | 360 |

13 | 68 | 75 | 156 | 390 |

34 | 55 | 87 | 176 | 396 |

11 | 90 | 97 | 198 | 396 |

13 | 109 | 120 | 242 | 396 |

20 | 51 | 65 | 136 | 408 |

24 | 37 | 37 | 98 | 420 |

25 | 34 | 39 | 98 | 420 |

29 | 29 | 40 | 98 | 420 |

29 | 29 | 42 | 100 | 420 |

21 | 41 | 50 | 112 | 420 |

26 | 35 | 51 | 112 | 420 |

25 | 39 | 56 | 120 | 420 |

14 | 61 | 65 | 120 | 420 |

37 | 37 | 70 | 144 | 420 |

26 | 51 | 73 | 150 | 420 |

41 | 50 | 89 | 180 | 420 |

26 | 73 | 97 | 196 | 420 |

21 | 85 | 104 | 210 | 420 |

15 | 106 | 119 | 240 | 420 |

7 | 169 | 174 | 350 | 420 |

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