In the figure, find the area of the shaded region. enclosed between two concentric circles of radii 7 cm and 14 cm where ∠AOC= 40 ° .

Given that the radius (r) of inner circle

= 7 cm .

Given that the radius (R) of the outer circle

= 14 cm .

Areas of sectors OAC and OBD are calculated as:
Angle made by the sector is given as

= 40 ° .

We know that,

Area of sector = \frac{θπ r^{2}}{36 0^{o}}

, where r is the radius of the circle.
Therefore,

Area of sector OAC = \frac{4 0^{o}}{36 0^{o}} \times π R^{2}

\Rightarrow Area of sector OAC = \frac{4 0^{o}}{36 0^{o}} \times \frac{22}{7} \times (14)^{2}

\Rightarrow Area of sector OAC = \frac{4 0^{o}}{36 0^{o}} \times \frac{22}{7} \times 196

\Rightarrow Area of sector OAC = \frac{1}{9} \times 22 \times 28

\Rightarrow Area of sector OAC = 68.44 c m^{2}

Area of sector OAC = \frac{4 0^{o}}{36 0^{o}} \times π R^{2}

\Rightarrow Area of sector OBD = \frac{4 0^{o}}{36 0^{o}} \times \frac{22}{7} \times (7)^{2}

\Rightarrow Area of sector OBD = \frac{4 0^{o}}{36 0^{o}} \times \frac{22}{7} \times 49

\Rightarrow Area of sector OBD = \frac{1}{9} \times 22 \times 7

\Rightarrow Area of sector OBD = 17.11 c m^{2}

From the figure,

Area of the small region ABCD = Area of the small sector OCD – Area of the small sector OBD

\Rightarrow Area of the small region ABCD = 68.44 - 17.11

\Rightarrow Area of the small region ABCD = 51.33 c m^{2}

From the figure,

Area of shaded region ABDC = Area of the outer circle - Area of inner circle - area of ABCD

We know that,

Area of circle = π r^{2}

, where r is the circle.

\Rightarrow Area of shaded region ABDC = π R^{2} - π r^{2} - 51.33

\Rightarrow Area of shaded region ABDC = π {(14)}^{2} - π {(7)}^{2} - 51.33

\Rightarrow Area of shaded region ABDC = π \times 196 - π \times 49 - 51.33

\Rightarrow Area of shaded region ABDC = \frac{22}{7} \times 196 - \frac{22}{7} \times 49 - 51.33

\Rightarrow Area of shaded region ABDC = 22 \times 28 - 22 \times 7 - 51.33

\Rightarrow Area of shaded region ABDC = 616 - 154 - 51.33

\Rightarrow Area of shaded region ABDC = 410.67 c m^{2}

The required area is equal to

410.67 c m 2 .

Hence, option 4 is the correct answer.

In the figure, find the area of the shaded region. enclosed between two concentric circles of radii $7 cm$ and $14 cm$ where $∠ AOC = 40 ° .$