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In the figure, PQRS is a square, O is center of the circle. If $RS = 102 c m 2$ , then find the area of shaded region.

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114 {cm}^{2}

128 {cm}^{2}

140 {cm}^{2}

116 {cm}^{2}

answer is A.

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Detailed Solution

Given that PQRS is a square and O is center of the circle.
We have to find the area of the shaded region if

RS = 102 c m 2

The expression to find the area of a semi-circular geometry is,

A = π r 2 2

, where r is the radius of a circle.
Since, PQRS is a square, then

P Q = Q R = P R = R S

. By using the Pythagoras theorem in the triangle PQR,
Question Image

Then, the radius, r of a semicircle is,
Question Image

Thus, the area of the shaded region,

A s

is,

A_{s} = area of circle - area of square

\Rightarrow A_{s} = π r^{2} - {(RS)}^{2}

\Rightarrow A_{s} = (3.14) {(10)}^{2} - {(10 \sqrt{2})}^{2}

\Rightarrow A_{s} = 314 - 200

\Rightarrow A_{s} = 114 {cm}^{2}

The area of the shaded region is

114 {cm}^{2}

.
Therefore, the correct option is 1.

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