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In figure, find the area of the shaded region, enclosed between two concentric circles of radii $7 cm$ and $14 cm$ where $∠ A O C = 40 ° ⋅$ $π = 227$

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420 c m^{2}

410.67 c m^{2}

None of these

431.12 c m^{2}

answer is A.

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Detailed Solution

Given that, Area of outer circle is 14 cm, area of inner circle is 7 cm and

∠ A O C

40 °

We know that, area of circle of radius r is equal to

π r 2

.
Area of sector which makes

θ

angle at the center of the circle of radius r is equal to

π r 2 θ 360 °

.
Let A

1

be the area of the outer circle.
Substitute 14 for r into

π r 2

to determine the area of the outer circle.

A 1 = π 142

⇒ 227 × 196

⇒

× 28

⇒

616
Let A

2

be the area of the inner circle.
Substitute 7 for r into

π r 2

to determine the area of the inner circle.

A 2 = π 72

⇒ 227 × 49

⇒ 22 × 7

⇒

154
Let

A_{G}

be the area of the gap between inner and outer circle.
Subtract the value of

A 2

from

A 1

to determine the area of the gap between inner and outer circle.

A G = A 1 − A 2

⇒

616

-

154

⇒

462
Subtract 14 for r and 40 ° for in the formula

π r 2 θ 360 °

and simplify to determine the area of the sector AOC .

a r e a A O C = π r 2 θ 360 °

⇒ 227 × 142 × 40 ° 360 °

⇒ 22 × 196 7 × 9

⇒

6169

Subtract 7 for r and 40° for

θ

in the formula

π r 2 θ 360 °

and simplify to determine the area of the sector BOD .

A r e a B O D = π r 2 θ 360 °

⇒ 227 × 72 × 40 ° 360 °

⇒ 22 × 49 7 × 9

⇒

1549

Subtract the area of sector AOC from the area of the gap between the inner and outer circles and add the area of the sector BOD to determine the area of the shaded region.

A = A G − A r e a A O C + A r e a B O D

⇒ 462 − 6169 + 1549

⇒ 462 − 4629

⇒ 462119

So,

A = 462 × 89

⇒

410.67
Therefore, the area of the shaded region is 410.67

m^{2}

.
Hence the correct option is 1.

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