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

The incentre of a triangle is the point of intersection of the angle bisectors of angles of the triangle. An incentre is also the centre of the circle touching all the sides of the triangle.

Note:
Angle bisector divides the oppsoite sides in the ratio of remaining sides i.e. BD/DC = AB/AC = c/b.

Incentre divides the angle bisectors in the ratio (b+c):a, (c+a):b and (a+b):c

Three angles of the triangle are

#A = pi/12, B = (3pi)/12, C = pi- pi/12 - (3pi) / 8 = (13pi)/24#

Area of triangle =3 = (bc Sin A)/2 = (ca sinB) / 2 = (ab sin C) /2#

#bc = (3 * 2) / sin (pi/12) = 23.1822#

#ca = (3 * 2) / sin ((3pi)/8) = 6.4944#

#ab = (3 * 2) / sin ((13pi)/24) = 6.0518#

#abc = sqrt(ab * bc * ca)#

#=>sqrt(23.1822 * 6.4944 * 6.0518) = 30.1863#

#a = (abc) / (bc) = 30.1863 / 23.1822 = 1.3021#

#b = (abc) / (ca) = 30.1863 / 6.4944 = 4.6481#

#c = (abc) / (ab) = 30.1863 / 6.0518 = 4.988#

Perimeter of the triangle #p = (a + b + c)#

#=> (1.3021 + 4.6481 + 4.988) = 10.9382#

The segments from the incenter to each vertex bisects each angle. The distances from the incenter to each side are equal to the inscribed circle's radius.

#r = 3 / (p/2) = 6 / ( * 10.9382) = 0.5485#

Area of incircle #A_i = pi r^2 = pi (0.5485)^2 = 1.7232#