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?

1 Answer
Douglas K.
Oct 17, 2016

#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 #

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