Question

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

• A. $66.5{\mathrm{cm}}^2$
• B. $66{\mathrm{cm}}^2$
• C. $65{\mathrm{cm}}^2$
• D. $67.5{\mathrm{cm}}^2$

Answer 1

The correct option is A

$66.5{\mathrm{cm}}^2$

The explanation for the correct answer.

Step 1: Find the radius and area of the circle with the diameter $\mathrm{OD}$

Given: Radius of the circle $\mathrm{OA}=7\mathrm{cm}=\mathrm{OD}.$

Radius of circle $=\frac{\mathrm{OD}}{2}=\frac{7}{2}\mathrm{cm}$

Area of the circle $={\mathrm{\pi r}}^2$

$$\begin{gathered} =\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2} \\ =38.5{\mathrm{cm}}^2 \end{gathered}$$

Step 2: Find the area of the semi-circle with a diameter of $\mathrm{AB}$

The radius of the semi-circle $=7\mathrm{cm}$

Area of the semi-circle $=\frac{1}{2}\times {\mathrm{\pi r}}^2$

$$\begin{gathered} =\frac{1}{2}\times \frac{22}{7}\times 7\times 7 \\ =77{\mathrm{cm}}^2 \end{gathered}$$

Step 3: Find the area of the triangle $\mathrm{BCA}$

The base of the triangle $\mathrm{AB}=7+7=14\mathrm{cm}$

Height of the triangle $\mathrm{OC}=\mathrm{OD}=7\mathrm{cm}$

Area of the triangle $=\frac{1}{2}\times \mathrm{b}\times \mathrm{h}$

$$\begin{gathered} =\frac{1}{2}\times 14\times 7 \\ =49{\mathrm{cm}}^2 \end{gathered}$$

Step 4: Area of the shaded region

Area of the shaded region $=$Area of the circle with diameter $\mathrm{OD}$ $+$Area of the semi-circle with diameter $\mathrm{AB}$ $-$Area of the triangle $\mathrm{ABC}$

Area of the shaded region $=38.5+77-49$

$=66.5{\mathrm{cm}}^2$

Hence option $A$ is the correct answer.