#"Area of "Delta = A_t = (1/2) a b sin C = (1/2) b c sin A = (1/2) c a sin B#
#"Given " hat A = pi/12, hat B = (5pi)/8, hat C = ((7pi)/24)#, A_t = 8#
#a b = (2 A_t) / sin C = 16 / sin ((7pi)/24) ~~ 20.1676#
#b c = (2 A_t) / sin A = 16 / sin (pi/12) = 61.8193#
#c a = (2 A_t) / sin B = 16 / sin ((5pi)/8) = 17.3183#
#a = (a b c) / (b c) = sqrt(20.1676 * 61.8193 * 17.3183) / 61.8193 = 2.3769#
#b = (a b c) / (c a) = sqrt(20.1676 * 61.8193 * 17.3183) / 17.3183 = 8.4847#
#c = (a b c) / (a b) = sqrt(2.3769 * 61.8193 * 17.3183) / 20.1676 = 7.286#
#"Semi-perimeter " = s = (a + b + c) / 2 = 18.1476 / 2 = 9.0738#
#"Radius of inscribed circle " = r = A_t / s = 8 / 9.0738#
#color(magenta)("Area of inscribed circle " = A_i = pi r^2 = pi * (8/9.0738)^2 ~~ 2.442#