Question

A triangle has two corners with angles of `pi / 12` and `pi / 6`. If one side of the triangle has a length of `16`, what is the largest possible area of the triangle?

Answer 1

Largest possible area of the triangle is `174.862`

Explanation:

As two angles of a triangle are `pi/6` and `pi/12`, the third angle is

`pi-pi/6-pi/12=(12pi-2pi-pi)/12=(9pi)/12`

All triangles with these angles are similar. As length of one side is `16`, its area will be maximum, if this side is opposite smallest angle i.e. `pi/12`.

Area of a triangle given one side `a=16` and two angles `/_A=pi/12`, `/_B=pi/6` and `/_C=(9pi)/12` is

`(a^2sinBsinC)/(2sinA)`

= `(16^2sin((9pi)/12)sin(pi/6))/(2sin(pi/12))`

= `(256xx0.7071xx1/2)/(2xx0.2588)`

= `(64xx0.7071)/0.2588`

= `174.862`