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 .
Answers
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