Home » Academics » Integer Triangles with a 120-Degree Angle

Integer Triangles with a 120° Angle

Just as there are integer right triangles, so there are integer 120° triangles. In such triangles, all side lengths are whole numbers and the obtuse angle measures exactly 120°.

If the two sides adjacent to the 120° angle are a and b and the third side is c, then a, b, and c satisfy a Pythagorean-like relation:

a² + ab + b² = c²,

which holds for all 120° triangles, not just the integer ones.

To find triples (a, b, c) that satisfy the relation a² + ab + b² = c², you can use generator functions similar to the generator functions for Pythagorean triples. In the 120° case, the equations are

a = m² - n²
b = 2mn + n²
c = m² + mn + n²,

where m and n are integers and m > n. Observe:

(m² - n²)² + (m² - n²)(2mn + n²) + (2mn + n²)² = (m² + mn + n²)².

In this way you can come up with infinitely many triples (a, b, c). The first few primitive triples (those in which a, b, and c have no factors in common) are given in the table below:

a	b	c
3	5	7
7	8	13
5	16	19
11	24	31
7	33	37
13	35	43
16	39	49
9	56	61
32	45	67
17	63	73
40	51	79
11	85	91
19	80	91
55	57	97

The first few triples with consecutive integer side lengths are

(7, 8, 13)
(104, 105, 181)
(1455, 1456, 2521)
(20272, 20273, 35113).