A triangle has vertices A, B, and C. Vertex A has an angle of pi/12 , vertex B has an angle of pi/6 , and the triangle's area is 15 . What is the area of the triangle's incircle?

1 Answer
Jan 3, 2018

The area of the triangle's incircle is 4.01 sq.unit.

Explanation:

/_A = pi/12= 180/12=15^0 , /_B = pi/6=180/6= 30^0

:. /_C= 180-(30+15)=135^0 : A_t=15

We know Area , A_t= 1/2*b*c*sinA or b*c=(2*15)/sin15 or

b*c ~~ 115.91, similarly , A_t= 1/2*a*c*sinB or

a*c=(2*15)/sin30 = 60 , and a*b=(2*15)/sin135 ~~ 42.43

(a*b)*(b*c)*(c.a)=(abc)^2= (42.43*115.91*60) or

abc=sqrt(42.43*115.91*60) ~~ 543.20(2dp)

a= (abc)/(bc)=543.20/115.91~~4.69

b= (abc)/(ac)=543.20/60~~9.05

c= (abc)/(ab)=543.20/42.43~~12.80

Semi perimeter : S/2=(4.69+9.05+12.80)/2 ~~13.27

Incircle radius is r_i= A_t/(S/2) = 15/13.27~~ 1.13

Incircla Area = A_i= pi* r_i^2= pi*1.13^2 ~~4.01 sq.unit

The area of the triangle's incircle is 4.01 sq.unit [Ans]