The following three shapes are based only on squares, semicircles, and quarter circles. Find the perimeter and the area of each shaded part. Give your answer as a completely simplified exact value in terms of π (no approximations).

Solution:- In the given figure Side of square= 10 cmarea of square =  side x side m= 10 x 10 = 100 sq cmNow , diameter of semi circle AOD = 2r = 10cm⇒ radius of  semi circle AOD = 5 cm∴ area of AOD = [tex]\frac{\pi\ r^2}{2}=\frac{\pi(5)^2}{2} =\pi \frac{25}{2}=12.5\pi cm^2[/tex]similarly Area of semicircle BOC = [tex]12.5\pi cm^2[/tex] Area of white portion in figure 1 = area of square - [ area of semicircle BOD + area of semicircle AOD]=[tex]=100-2(12.5\pi) =100-25\pi[/tex]Total area of white portion in the given figure=[tex]100-25\pi[/tex]∴Area of Shaded portion = area of square - area of total white portion = [tex]100-(200-50\pi )=100-200+50\pi =50\pi -100\ cm^2[/tex]So the area of shaded portion is [tex]50\pi -100\ cm^2[/tex]Now perimeter of AD(which is a semicircle) = [tex]\pi\ r=3.14\cdot\ 5=15.7cm[/tex](as perimeter of semicircle=πr)So the perimeter of shaded region = 4×perimeter of AD=4×15.7=62.8cm