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Find the area of the segment shown in figure, if radius of the circle is $21 cm$ and $∠ A O B = 120 °$ .

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271.04

{cm}^{2}

275

{cm}^{2}

280

{cm}^{2}

None of these

answer is A.

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Detailed Solution

Given that, radius of the circle is 21 cm and angle AOB is 120 °.
Question Image

We know that, area of sector which makes

θ

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

π 2 θ 360 °

.
Also we know that, area of the triangle made by the chord of the circle which makes an angle at the center of the circle is equal to

12 r 2 sin θ

.
Determine the area of the sector of the circle by using the formula

A 1 = q 360 ° π r 2

A 1 = 120 360 ° × 227212

\Rightarrow

462

{cm}^{2}

Find the area of the triangle AOB by using the formula

12 r 2 sin θ

A 2 = 12212 sin 120 °

\Rightarrow

190.3

{cm}^{2}

The area of the segment S represents the difference between the area

A 2

of triangle ABO and area of the sector of the circle

A 1

.
Subtracting the value of

A 2

from

A 1

to determine the value of S,

S = A 1 − A 2

⇒ 462 − 190.9

⇒

271.04

{cm}^{2}

Therefore, area of the segment is 271.04

{cm}^{2}

.
Hence the correct option is 1.

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