Question

In Fig., $ 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 $ 0A=7 cm$ find the area of the shaded region.

Answer 1

Step 1: Find the area of shaded circle

From the given condition the radius $0A=7cm$

$OD$ is the diameter of the shaded circle and also the radius of the big circle .

$\therefore OD=7cm$

We know that area of circle

$=\pi r^2\left[\mathrm{r}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{circle}\right]$

Therefore the area of the circle whose diameter is $OD$ is

$$\begin{gathered} =\mathrm{\pi }·{\left(\frac{OD}{2}\right)}^2 \\ =\mathrm{\pi }·{\left(\frac{7}{2}\right)}^2 \\ =\frac{22}{7}·\frac{49}{4} \\ =38.5{\mathrm{cm}}^2 \end{gathered}$$

Step 2: Find the area of shaded arc

Area of the triangle whose base is $AB$ and height is $OC$ is

Area of triangle

$$\begin{gathered} =\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height} \\ =\frac{1}{2}\times \mathrm{AB}\times \mathrm{OC} \\ =\frac{1}{2}\times 14\times 7\left[\because \mathrm{AB}=2\mathrm{OA}=14\mathrm{cm}\mathrm{and}\mathrm{OC}=\mathrm{OA}=7\mathrm{cm}\right] \\ =49{\mathrm{cm}}^2 \end{gathered}$$

Area of the half circle whose radius is $OC$ IS

Area of half =circle$=\pi r^2\left[\mathrm{r}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{circle}\right]$

$$\begin{gathered} =\frac{\mathrm{\pi }}{2}{\left(7\right)}^2 \\ =\frac{22}{7\times 2}\times 49 \\ =77c{m}^2 \end{gathered}$$

Therefore the area of the shaded arc is

$$\begin{gathered} =\mathrm{Area}o\mathrm{f}\mathrm{half}\mathrm{circle}-\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle} \\ =77-49 \\ =28c{m}^2 \end{gathered}$$

Step 3: Find the area of shaded region.

The area of the shaded region is

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{arc}+\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{circle} \\ =28+38.5 \\ =66.5c{m}^2 \end{gathered}$$

Hence the area of the shaded region is $66.5c{m}^2$