Question

In the given figure, AB and CD are two diameters of a circle perpendicular to each other and OD is the diameter of the smaller circle. If OA=7 , find the area of the shaded region.

Answer 1

Hint: To find the area of the shaded region in the given figure, it can be calculated by subtracting the area of the small circle to the area of the bigger circle. For this calculation of radius of respective circles correctly holds the key.

Complete step-by-step answer:
 Radius of big circle \[\left( R \right) = 7\] cm
∴ Area of big circle \[ = \pi {R^2}\]
On putting the value of radius
⇒ Area of big circle \[ = \dfrac{{22}}{7} \times {\left( 7 \right)^2}\]
∴ Area of big circle \[ = 154\] $cm^2$ -- ( 1 )
now,
⇒ Radius of small circle \[\left( r \right) = \dfrac{7}{2} = 3.5\] cm
Area of smaller circle \[ = \pi {r^2}\]
⇒ Area of small circle \[ = \dfrac{{22}}{7} \times {\left( {3.5} \right)^2}\]
∴ Area of small circle \[ = 38.5\] $cm^2$ -- ( 2 )
⇒ Area of shaded region = Area of big circle - Area of small circle.
on putting the value of area of both the circles on the above formula we get.
⇒ Area of shaded region \[ = 154 - 38.5 = 115.5\] $cm^2$
Hence the required area of the shaded region is
So, the correct answer is “115.5 $cm^2$”.

Note: We can also calculate the area of the shaded region by calculating first the area of the semi circle of the bigger circle then subtract the area of the smaller circle from the area of the semicircle calculated previously. In the final the area of the shaded region will be the equal to the area of the bigger semicircle plus area of another bigger semicircle minus the area of the smaller circle.