Question

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).

Answer 1

Solution:- In the given figure

Side of square= 10 cm

area of square =  side x side m= 10 x 10 = 100 sq cm

Now , diameter of semi circle AOD = 2r = 10cm

⇒ radius of  semi circle AOD = 5 cm

∴ area of AOD = $\frac{\pi\ r^2}{2}=\frac{\pi(5)^2}{2} =\pi \frac{25}{2}=12.5\pi cm^2$

similarly Area of semicircle BOC = $12.5\pi cm^2$

Area of white portion in figure 1 = area of square - [ area of semicircle BOD + area of semicircle AOD]

=$=100-2(12.5\pi) =100-25\pi$

Total area of white portion in the given figure=$100-25\pi$

∴Area of Shaded portion = area of square - area of total white portion

= $100-(200-50\pi )=100-200+50\pi =50\pi -100\ cm^2$

So the area of shaded portion is $50\pi -100\ cm^2$

Now perimeter of AD(which is a semicircle) = $\pi\ r=3.14\cdot\ 5=15.7cm$(as perimeter of semicircle=πr)

So the perimeter of shaded region = 4×perimeter of AD=4×15.7=62.8cm