Question

QUESTION 8b: The following three shapes are based only on squares, semicircles, and quarter circles. Find the perimeter AND area of each shaded part. Give your answer as a COMPLETELY simplified exact value in terms of π (no approximations).

Please give your answer with an explanation. An image of your work on paper would be greatly appreciated .

Answer 1

Step-by-step explanation:

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

2

π r

2

=

2

π(5)

2

=π

2

25

=12.5πcm

2

similarly Area of semicircle BOC = 12.5\pi cm^212.5πcm

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=100−2(12.5π)=100−25π

Total area of white portion in the given figure=100-25\pi100−25π

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

= 100-(200-50\pi )=100-200+50\pi =50\pi -100\ cm^2100−(200−50π)=100−200+50π=50π−100 cm

2

So the area of shaded portion is 50\pi -100\ cm^250π−100 cm

2

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

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