A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/3 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?

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Answer 1 · 2018-04-15T12:30:13

#0.42 units^2#

The formula for the radius, #r# of incircle is given by #r=A_t/s#, where #A_t=#Area of the Triangle, and #s=#semi-perimeter.

#triangleABC# is a right-angled triangle since vertex A#=pi/2#.

Since vertex B #=pi/3#, using trigonometric ratios, #a=2, b=sqrt(3), c=1#.

#r=A_t/s=(1/2(1)(sqrt3))/(1/2(1+2+sqrt3))=sqrt(3)/(3+sqrt(3))=(sqrt3-1)/2#

Hence, area of incircle #=pir^2=pi((sqrt3-1)/2)^2approx0.42 units^2#