# Heronian triangle

In geometry, a **Heronian triangle** is a triangle that has side lengths and area that are all integers.^{[1]}^{[2]} Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers.^{[3]}

## Contents

## Properties

Any right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the product of the two shorter sides of the triangle, at least one of which must be even.

An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the picture to the right. One takes a Pythagorean triple (*a*, *b*, *c*), with *c* being largest, then another one (*a*, *d*, *e*), with *e* being largest, constructs the triangles with these sidelengths, and joins them together along the sides of length *a*, to obtain a triangle with integer side lengths *c*, *e*, and *b* + *d*, and with area

- (one half times the base times the height).

If *a* is even then the area *A* is an integer. Less obviously, if *a* is odd then *A* is still an integer, as *b* and *d* must both be even, making *b*+*d* even too.

Some Heronian triangles cannot be obtained by joining together two right-angled triangles with integer sides as described above. For example a 5, 29, 30 Heronian triangle with area 72 cannot be constructed from two integer Pythagorean triangles since none of its altitudes are integers. Also no primitive Pythagorean triangle can be constructed from two smaller integer Pythagorean triangles.^{[4]}^{:p.17} Such Heronian triangles are known as **indecomposable**.^{[4]} However, if one allows Pythagorean triples with rational values, not necessarily integers, then a decomposition into right triangles with rational sides always exists,^{[5]} because every altitude of a Heronian triangle is rational (since it equals twice the integer area divided by the integer base). So the Heronian triangle with sides 5, 29, 30 can be constructed from rational Pythagorean triangles with sides 7/5, 24/5, 5 and 143/5, 24/5, 29. Note that a Pythagorean triple with rational values is just a scaled version of a triple with integer values.

Other properties of Heronian triangles are given in Integer triangle#Heronian triangles.

## Exact formula for Heronian triangles

Every Heronian triangle has sides proportional to:^{[6]}

for integers *m*, *n* and *k* where:

- .

The proportionality factor is generally a rational where reduces the generated Heronian triangle to its primitive and scales up this primitive to the required size. For example, taking *m* = 36, *n* = 4 and *k* = 3 produces a triangle with *a* = 5220, *b* = 900 and *c* = 5400, which is similar to the 5, 29, 30 Heronian triangle and the proportionality factor used has *p* = 1 and *q* = 180.

See also Heronian triangles with one angle equal to twice another, Heronian triangles with sides in arithmetic progression, and Isosceles Heronian triangles.

## Examples

The list of primitive integer Heronian triangles, sorted by area and, if this is the same, by perimeter, starts as in the following table. "Primitive" means that the greatest common divisor of the three side lengths equals 1.

Area | Perimeter | side length b+d | side length e | side length c |
---|---|---|---|---|

6 | 12 | 5 | 4 | 3 |

12 | 16 | 6 | 5 | 5 |

12 | 18 | 8 | 5 | 5 |

24 | 32 | 15 | 13 | 4 |

30 | 30 | 13 | 12 | 5 |

36 | 36 | 17 | 10 | 9 |

36 | 54 | 26 | 25 | 3 |

42 | 42 | 20 | 15 | 7 |

60 | 36 | 13 | 13 | 10 |

60 | 40 | 17 | 15 | 8 |

60 | 50 | 24 | 13 | 13 |

60 | 60 | 29 | 25 | 6 |

66 | 44 | 20 | 13 | 11 |

72 | 64 | 30 | 29 | 5 |

84 | 42 | 15 | 14 | 13 |

84 | 48 | 21 | 17 | 10 |

84 | 56 | 25 | 24 | 7 |

84 | 72 | 35 | 29 | 8 |

90 | 54 | 25 | 17 | 12 |

90 | 108 | 53 | 51 | 4 |

114 | 76 | 37 | 20 | 19 |

120 | 50 | 17 | 17 | 16 |

120 | 64 | 30 | 17 | 17 |

120 | 80 | 39 | 25 | 16 |

126 | 54 | 21 | 20 | 13 |

126 | 84 | 41 | 28 | 15 |

126 | 108 | 52 | 51 | 5 |

132 | 66 | 30 | 25 | 11 |

156 | 78 | 37 | 26 | 15 |

156 | 104 | 51 | 40 | 13 |

168 | 64 | 25 | 25 | 14 |

168 | 84 | 39 | 35 | 10 |

168 | 98 | 48 | 25 | 25 |

180 | 80 | 37 | 30 | 13 |

180 | 90 | 41 | 40 | 9 |

198 | 132 | 65 | 55 | 12 |

204 | 68 | 26 | 25 | 17 |

210 | 70 | 29 | 21 | 20 |

210 | 70 | 28 | 25 | 17 |

210 | 84 | 39 | 28 | 17 |

210 | 84 | 37 | 35 | 12 |

210 | 140 | 68 | 65 | 7 |

210 | 300 | 149 | 148 | 3 |

216 | 162 | 80 | 73 | 9 |

234 | 108 | 52 | 41 | 15 |

240 | 90 | 40 | 37 | 13 |

252 | 84 | 35 | 34 | 15 |

252 | 98 | 45 | 40 | 13 |

252 | 144 | 70 | 65 | 9 |

264 | 96 | 44 | 37 | 15 |

264 | 132 | 65 | 34 | 33 |

270 | 108 | 52 | 29 | 27 |

288 | 162 | 80 | 65 | 17 |

300 | 150 | 74 | 51 | 25 |

300 | 250 | 123 | 122 | 5 |

306 | 108 | 51 | 37 | 20 |

330 | 100 | 44 | 39 | 17 |

330 | 110 | 52 | 33 | 25 |

330 | 132 | 61 | 60 | 11 |

330 | 220 | 109 | 100 | 11 |

336 | 98 | 41 | 40 | 17 |

336 | 112 | 53 | 35 | 24 |

336 | 128 | 61 | 52 | 15 |

336 | 392 | 195 | 193 | 4 |

360 | 90 | 36 | 29 | 25 |

360 | 100 | 41 | 41 | 18 |

360 | 162 | 80 | 41 | 41 |

390 | 156 | 75 | 68 | 13 |

396 | 176 | 87 | 55 | 34 |

396 | 198 | 97 | 90 | 11 |

396 | 242 | 120 | 109 | 13 |

## Equable triangles

A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17).^{[7]}^{[8]}

## Almost-equilateral Heronian triangles

Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form *n* − 1, *n*, *n* + 1. A method for generating all solutions to this problem based on continued fractions was described in 1864 by Edward Sang,^{[9]} and in 1880 Reinhold Hoppe gave a closed-form expression for the solutions.^{[10]} The first few examples of these almost-equilateral triangles are listed in the following table (sequence A003500 in OEIS):

Side length | Area | Inradius | ||
---|---|---|---|---|

n − 1 |
n |
n + 1 |
||

3 | 4 | 5 | 6 | 1 |

13 | 14 | 15 | 84 | 4 |

51 | 52 | 53 | 1170 | 15 |

193 | 194 | 195 | 16296 | 56 |

723 | 724 | 725 | 226974 | 209 |

2701 | 2702 | 2703 | 3161340 | 780 |

10083 | 10084 | 10085 | 44031786 | 2911 |

37633 | 37634 | 37635 | 613283664 | 10864 |

Subsequent values of *n* can be found by multiplying the previous value by 4, then subtracting the value prior to that one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc.), thus:

where *t* denotes any row in the table. This is a Lucas sequence. Alternatively, the formula generates all *n*. Equivalently, let *A* = area and *y* = inradius, then,

where {*n*, *y*} are solutions to *n*^{2} − 12*y*^{2} = 4. A small transformation *n* = *2x* yields a conventional Pell equation *x*^{2} − 3*y*^{2} = 1, the solutions of which can then be derived from the regular continued fraction expansion for √3.^{[11]}

The variable *n* is of the form , where *k* is 7, 97, 1351, 18817, …. The numbers in this sequence have the property that *k* consecutive integers have integral standard deviation.^{[12]}

## See also

## External links

- Weisstein, Eric W., "Heronian triangle",
*MathWorld*. - Online Encyclopedia of Integer Sequences Heronian
- Wm. Fitch Cheney, Jr. (January 1929), "Heronian Triangles",
*Am. Math. Monthly*,**36**(1): 22–28, JSTOR 2300173<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - S. sh. Kozhegel'dinov (1994), "On fundamental Heronian triangles",
*Math. Notes*,**55**(2): 151–6, doi:10.1007/BF02113294<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## References

- ↑ Carlson, John R. (1970), "Determination of Heronian Triangles" (PDF),
*Fibonacci Quarterly*,**8**: 499–506<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Beauregard, Raymond A.; Suryanarayan, E. R. (January 1998), "The Brahmagupta Triangles" (PDF),
*College Math Journal*,**29**(1): 13–17, doi:10.2307/2687630<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W., "Heronian Triangle",
*MathWorld*. - ↑
^{4.0}^{4.1}Yiu, Paul (2008),*Heron triangles which cannot be decomposed into two integer right triangles*(PDF), 41st Meeting of Florida Section of Mathematical Association of America<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Sierpiński, Wacław (2003) [1962],
*Pythagorean Triangles*, Dover Publications, Inc., ISBN 978-0-486-43278-6<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Carmichael, R. D., 1914, "Diophantine Analysis", pp.11-13; in R. D. Carmichael, 1959,
*The Theory of Numbers and Diophantine Analysis*, Dover Publications, Inc. - ↑ Dickson, Leonard Eugene (2005),
*History of the Theory of Numbers, Volume Il: Diophantine Analysis*, Dover Publications, p. 199, ISBN 9780486442334<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Markowitz, L. (1981), "Area = Perimeter",
*The Mathematics Teacher*,**74**(3): 222–3, Zbl 1982d.06561 Check`|zbl=`

value (help)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Sang, Edward, "On the theory of commensurables",
*Transactions of the Royal Society of Edinburgh*,**23**: 721–760<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. See in particular p. 734. - ↑ Gould, H. W. (February 1973), "A triangle with integral sides and area" (PDF),
*Fibonacci Quarterly*,**11**(1): 27–39<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. - ↑ Richardson, William H. (2007),
*Super-Heronian Triangles*<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Online Encyclopedia of Integer Sequences, A011943.