#hat A = pi/12, hat B = pi/2, hat C = (5pi) / 12, A_t = 24#
#A_t = (1/2) a b sin C = (1/2) bc sin A = (1/2) ca sin B#
#ab = (2 A_t) / sin C = 48 / sin ((5pi)/12) = 49.69#
#bc = A_t / sin A = 48/ sin (pi/12) = 185.46#
#ca = A_t / sin C = 48 / sin(pi/2) = 48#
#a = (abc) / (bc) = sqrt(185.46 * 48 * 49.69) / 185.46 = 3.59#
#b = (abc) / (ac) = sqrt(185.46 * 48 * 49.69) / 48 =13.86#
#c = (abc) / (ab) = sqrt(185.46 * 48 * 49.69) / 49.69 = 13.84#
#"Semiperimeter " = s = (a + b + c) / 2 = (3.59 + 13.86 + 13.84) / 2 = 15.645#
#"Incircle radius " = r = A_t / s = 24 / 15.645#
#color(purple)("Area of incircle " = A_i = pi * (24/15.645)^2 = 7.393 " sq units"#