Question

In the figure below, AB and CD are two diameters of a circle (with center O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

• A. 65.5 $c{m}^2$
• B. 66.5 $c{m}^2$
• C. 67.5 $c{m}^2$
• D. 68.5 $c{m}^2$

Answer 1

The correct option is B 66.5 $c{m}^2$

Radius of larger circle, R = 7 cm
Radius of smaller circle, $r=\frac{7}{2}cm$
Height of $\Delta BCA=OC=7cm$
Base of $\Delta BCA=AB=14cm$
Area of $\Delta BCA=\frac{1}{2}\times AB\times OC=\frac{1}{2}\times 7\times 14=49c{m}^2$
Area of larger circle $=\pi R^2=\frac{22}{7}\times 7^2=154c{m}^2$
Area of larger semicircle $=\frac{154}{2}c{m}^2=77c{m}^2$
Area of smaller circle $=\pi r^2=\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}=\frac{77}{2}c{m}^2$
Area of the shaded region = Area of larger circle - Area of triangle - Area of larger semicircle + Area of smaller circle
Area of the shaded region $=(154-49-77+\frac{77}{2})c{m}^2$
$=\frac{133}{2}c{m}^2=66.5c{m}^2$