Question

A triangle has two corners with angles of `pi / 12` and `pi / 8`. If one side of the triangle has a length of `6`, what is the largest possible area of the triangle?

Answer 1

Largest possible area of triangle = 262.5667

Explanation:

Three angles are `pi/12, pi/8, (17pi)/24`

To get larges area, length 6 must correspond to least angle `pi/12`

`6/ sin (pi/12) = b / sin (pi/8) = c / sin ((19pi)/24)`

`b = (6 * sin (pi)/8) / sin (pi/12) = 8.8715`

`c = (6*sin ((19pi)/24)) / sin (pi/ 12) = 14.1124`

Semiperimeter `s = ( a + b + c)/2 = (6+8.8715+14.1124)/2 = 14.492`

`s-a = 14.492 - 6 = 8.492`
`s-b = 14.492 - 8.8715 = 5.6205`
`s-c = 14.492 - 14.1124 = 0.3796`

Area of `Delta = sqrt(s (s-a) (s-b) (s-c))`

Area of `Delta = sqrt(14.492 * 8.492 * 5.6205 * 0.3796) = **262.5667**`