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In figure, two concentric circles with centre O, have radii $21 cm$ and $42 cm .$ If $∠ A O B = 60 °$ find the area of the shaded region.

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3400 c m^{2}

3450 c m^{2}

3500 c m^{2}

3465 c m^{2}

answer is A.

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Detailed Solution

Given that, two concentric circles with centre O, have radii 21 cm and 42 cm and

∠ A O B − 60

°.

We know that, the area of sector is

θ 360 × π r 2

, where

θ

is the angle subtended at the centre of the circle.
We know that, the area of circle is

π r 2

.
Calculating the area of the region ABCD, subtract the area of the sector COD from the area of the sector AOB.

θ 360 × π r 12 − θ 360 × π r 22

⇒ 60360 × 227 × 242 − 60360 × 227 × 212

⇒ 16 × 22 × 6 × 42 − 16 × 22 × 3 × 21

⇒ 22 × 42 − 11 × 21 c m 2

⇒ 924 − 231 c m 2

⇒

693

{cm}^{2}

Calculating the difference of the area of the bigger circular ring and the smaller circular ring.

π r 12 − π r 22

⇒ 227 × 42 × 42 − 227 × 21 × 21

⇒ 22 × 6 × 42 − 22 × 3 × 21

⇒

5544 - 1386

⇒ 4158 c m 2

To find the required shaded region, subtract the area of region ABCD from the area of circular ring.

⇒

4158

-

693

⇒

3465
The area of the shaded region is

3465 c m 2

.
Hence the correct option is 1.

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