The figures below are made out of circles, semicircles, quarter circles, and a square. Find the area and the perimeter of each figure and give your answers as a completely simplified exact value in terms of π (no approximations).

Accepted Solution

Answer:Area of the shaded region = 16(π - 2) in²Perimeter of the shaded region = 4(π + [tex]2\sqrt{2}[/tex]) inStep-by-step explanation:Since, BDC is a quarter of the circle with radius = 8 in.Area of the quarter of the circle = [tex]\frac{1}{4}(\pi)(r)^2[/tex]                                                      = [tex]\frac{1}{4}(64)\pi[/tex]                                                      = 16π in²Area of ΔBCD = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]                         = [tex]\frac{1}{2}(8)(8)[/tex]                         = 32 in²Since, area of the shaded part = Area of quarter of the circle - Area of triangle BCD= (16π - 32)= 16(π - 2) in²Therefore, area of the shade region = 16(π - 2) in²Similarly, length of arc BD = [tex]\frac{1}{4}(2\pi r)[/tex]                                             = [tex]\frac{8\pi }{2}[/tex]                                             = 4π in.Length of the diagonal of a square = (Side)√2                                                           = 8√2 inPerimeter of the shaded region = Length of arc BD + Length of diagonal BD= 4π + 8√2= 4(π + 2√2) in.