Question

$ AB$ and $ CD$ are arcs of two concentric circles of radii $ \text{21 cm}$ and $ \text{7 cm}$ respectively and centre at $ O$. If $ \text{∠AOB = 30°}$, find the area of the shaded region.

s Tem 21em 

Answer 1

Step $1:$ Finding the areas of sectors OCD and OAB:

Let ${\text{A}}_1$ and ${\text{A}}_2$ be the areas of sectors OCD and OAB respectively.

Angle made by both sectors at the centre of circle $\text{= 30°}$.

${\text{A}}_1\text{=}$ Area of sector OCD whose radius is $\text{7 cm}$, is

$\begin{array}{rcl}& & {\text{A}}_1\mathrm{=}\frac{30}{360}\mathrm{\times }\frac{22}{7}\times 7^2\mathbf{}\left(\mathrm{Using}A=\frac{\theta }{360}\times {\mathrm{\pi r}}^2\right)\\ & & \mathrm{=}\frac{77}{6}{\mathrm{cm}}^2\end{array}$

${\text{A}}_2\text{=}$ Area of sector OAB whose radius is $\text{21 cm}$, is

$\begin{array}{rcl}& & {\text{A}}_2\mathrm{=}\frac{30}{360}\mathrm{\times }\frac{22}{7}\times {21}^2\mathbf{}\mathbf{(}\mathbf{Using}\mathbf{}\mathbf{A}\mathbf{=}\frac{\mathbf{\theta }}{\mathbf{360}}\mathbf{\times }{\mathbf{\pi r}}^{\mathbf{2}}\mathbf{)}\\ & & \mathrm{=}\frac{462}{4}{\mathrm{cm}}^2\end{array}$

Step $2:$ Finding the shaded area:

$$\begin{gathered} \text{Area of shaded region=Area of sector OAB-Area of sector OCD} \\ {\text{=A}}_2-{\mathrm{A}}_1 \\ =\frac{462}{4}-\frac{77}{6} \\ =\frac{462\times 3-77\times 2}{12} \\ =\frac{1386-154}{12} \\ =\frac{1232}{12} \\ =102.67{\mathrm{cm}}^2 \end{gathered}$$

Hence, the area of shaded region is ${\text{102.67 cm}}^2$.