Question

Two corners of a triangle have angles of `pi / 8` and `pi / 6`. If one side of the triangle has a length of `4`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter is `17.519`.

Explanation:

As two angles are `pi/8` and `pi/6`, third angle is `pi-pi/8-pi/6=(24pi-3pi-4pi)/24-(17pi)/24`.

For longest perimeter side of length `4`, say `a`, has to be opposite smallest angle `pi/8` and then using sine formula other two sides will be

`4/(sin(pi/8))=b/(sin(pi/6))=c/(sin((17pi)/24))`

Hence `b=(4sin(pi/6))/(sin(pi/8))=(4xx0.5)/0.3827=5.226`

and `c=(4xxsin((17pi)/24))/(sin(pi/8))=(4xx0.7934)/0.3827=8.293`

Hence longest perimeter is `4+5.226+8.293=17.519`.