The distances from the incenter to each side are equal to the inscribed circle's radius. The area of the triangle is equal to #color(crimson)("12×r×(the triangle's perimeter), 1 2 × r × ( the triangle's perimeter ) "#, where r is the inscribed circle's radius.
#"Area of Triangle " A_t = (ab sin C) / 2 = (bc sin A) / 2 = (ca sin B) / 2#
#hat A = pi/8, hat b = pi/3, hat C = (13pi)/24, A_t = 12#
#ab = (2 * A_t) / sin C = (2 * 12) / sin ((13pi)/24) = 24.21#
#bc = (2 * A_t) / sin A = (2 * 12) / sin (pi/8) = 62.72#
#ca = (2 * A_t) / sin B = (2 * 12) / sin (pi/3) = 27.71#
#a = sqrt(ab * bc * ca) / (bc) = sqrt(24.21 * 62.72 * 27.71) / 62.72 = 3.27#
#b = sqrt(ab * bc * ca) / (bc) = sqrt(24.21 * 62.72 * 27.71) / 27.71 = 7.4#
#c = sqrt(ab * bc * ca) / (ab) = sqrt(24.21 * 62.72 * 27.71) / 24.21 = 8.47#
#"Perimeter " p = a + b + c = 3.27 + 7.4 + 8.47 = 19.14#
#Incircle radius " r = A_t / (12 * p) = 12 / (12 * 19.14) = 0.0522#
#color(brown)("Area of inscribed circle " A_r = pi r^2 = pi * 0.0522^2 = 0.0086 " sq units"#
