Question

Area of the shaded portion in given figure is

A) $7.5 \pi$ square units
B) $6.5 \pi$ square units
C) $5.5 \pi$ square units
D) $4.5 \pi$ square units

Answer 1

Hint:
We can observe two semicircles with the same center but different radii. We have to find the area of the shaded portion. For that we have to remove the area of the inner semi circle from the area of outer semicircle.
Area of semicircle = \[\dfrac{{\pi {r^2}}}{2}\]

Complete step by step solution:
Let the radius of inner semicircle be $r = 5-1=4$ units

Radius of outer circle be R = 5 units
Now area of shaded portion
\[ \Rightarrow A\left( {outer{\text{ }}semicircle} \right) - A\left( {inner{\text{ }}semicircle} \right)\]
\[ \Rightarrow \dfrac{{\pi {R^2}}}{2} - \dfrac{{\pi {r^2}}}{2}\]
Taking \[\dfrac{\pi }{2}\] common
\[ \Rightarrow \dfrac{\pi }{2}\left( {{R^2} - {r^2}} \right)\]
Substitute the values of radius
\[
   \Rightarrow \dfrac{\pi }{2}\left( {{5^2} - {4^2}} \right) \\
   \Rightarrow \dfrac{\pi }{2}\left( {25 - 16} \right) \\
   \Rightarrow \dfrac{{9\pi }}{2} \\
   \Rightarrow 4.5\pi sq.units \\
\]

Hence the area of the shaded portion is \[ \Rightarrow 4.5 \pi sq.units\].
So option D is correct.

Note:
In this problem generally students get confused in taking the correct radius. Here 5units is the radius of the outer semicircle and 1unit is the length of that shaded border. So to find the radius of the inner semicircle we have to subtract that border from the radius of outer semicircle.