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Find the area of the shaded region in figure, where arcs drawn with centres A,B,C and $D$ intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side of $12 cm$ .

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28 c m^{2}

30.96 c m^{2}

20 c m^{2}

32 c m^{2}

answer is A.

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Detailed Solution

Given that, the side of the square A, B, C and D is 12 cm.
Question Image

The shaded region represents the difference between area of square and area of the shaded regions which area 4 in count.
Determining the area of square

A_{1}

by using the formula for area of square

A 1 = s i d e 2

A 1 = 122

\Rightarrow

144

c m^{2}

Determining the area for a quadrant A of the square by using the formula for area of quadrant

A = p r 2 4

in which r represents the radius that is half of side of the square,

A = 3.14 62 4

⇒ 3.14 × 36 4

⇒ 113.04 4

\Rightarrow

28.26

c m^{2}

Determining the area of all 4 quadrants

A_{2}

A_{2} = 28.26 \times 4

\Rightarrow

113.04

c m^{2}

Determining the shaded region S by subtracting the value of

A_{1}

from

A_{2}

we get,

S = A 1 − A 2

⇒ 144 − 113.04

\Rightarrow

30.96

c m^{2}

Therefore, area represents the shaded region is 30.96

c m^{2}

.
Hence the correct option is 1.

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