Question

A triangle has vertices A, B, and C. Vertex A has an angle of `pi/8`, vertex B has an angle of `( pi)/6`, and the triangle's area is `32`. What is the area of the triangle's incircle?

Answer 1

`A_i ~~ 10.865`

Explanation:

Incircle of a Triangle

Let `Delta` be the area of the triangle `= 32`
Let `/_A = pi/8`
Let `/_B = pi/6`
Let `/_C = pi - pi/6 - pi/8 = (48pi)/48 - (8pi)/48 - (6pi)/48 = (17pi)/24`
Let `r` be the radius of the incircle.

Using an equation from the reference page, we can write the following:

`r^2 = Delta/(cot(A/2) + cot(B/2) + cot(C/2)`

Multiply by `pi` to obtain the area of the incircle:

`A_(i) = (Deltapi)/(cot(A/2) + cot(B/2) + cot(C/2)`

`A_i ~~ 10.865`