Question

A triangle has vertices A, B, and C. Vertex A has an angle of `pi/12`, vertex B has an angle of `(3pi)/4`, and the triangle's area is `8`. What is the area of the triangle's incircle?

Answer 1

The area of the incircle is `=2.14u^2`

Explanation:

The area of the triangle is `A=8`

The angle `hatA=1/12pi`

The angle `hatB=3/4pi`

The angle `hatC=pi-(1/12pi+3/4pi)=1/6pi`

The sine rule is

`a/(sin hat (A))=b/sin hat (B)=c/sin hat (C)=k`

So,

`a=ksin hatA`

`b=ksin hatB`

`c=ksin hatC`

Let the height of the triangle be `=h` from the vertex `A` to the opposite side `BC`

The area of the triangle is

`A=1/2a*h`

But,

`h=csin hatB`

So,

`A=1/2ksin hatA*csin hatB=1/2ksin hatA*ksin hatC*sin hatB`

`A=1/2k^2*sin hatA*sin hatB*sin hatC`

`k^2=(2A)/(sin hatA*sin hatB*sin hatC)`

`k=sqrt((2A)/(sin hatA*sin hatB*sin hatC))`

`=sqrt(16/(sin(1/12pi)*sin(3/4pi)*sin(1/6pi)))`

`=13.22`

Therefore,

`a=13.22sin(1/12pi)=3.42`

`b=13.22sin(3/4pi)=9.35`

`c=13.22sin(1/6pi)=6.61`

The radius of the incircle is `=r`

`1/2*r*(a+b+c)=A`

`r=(2A)/(a+b+c)`

`=16/(19.38)=0.83`

The area of the incircle is

`area=pi*r^2=pi*0.83^2=2.14u^2`