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From each corner of a square of side $4 cm$ , a quadrant of a circle of radius $1 cm$ is cut, and also a circle of diameter $2 cm$ is cut as shown in the figure. The area of the remaining (shaded) portion is

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(16 − 2 π) cm 2

(16 − 5 π) cm 2

2 π cm 2

5 π cm 2

answer is A.

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Detailed Solution

It is given that from each corner of a square of side

4 cm

, a quadrant of a circle of radius

1 cm

is cut, and also a circle of diameter

2 cm

is cut.

The area of the circle is given by

A = π r 2,

where r is the radius of the circle.
Side of the square is

a = 4 cm

.
So, the area of a square ABCD is,

A = a 2

⇒

A = 42

⇒

A = 16 cm 2

It is given that the diameter of the circle is

d = 2 cm .

Thus, radius = 1cm.
The area of a middle circle is,

⇒ A' = π (1) 2 ⇒ A' = π

It is given that the radius of the quadrant of a circle is

r = 1 cm

.
So the area of 4 quadrants are,

A c = 4 × π r 2 90 ° 360 °

⇒ A c = 4 × π (1) 2 4 ⇒ A c = π

The area of the remaining portion of the square is,

A s = A − A c + A'

⇒ A s = 16 − (π + π)

⇒ A s = (16 − 2 π) cm 2

Hence, the area of shaded region is (

16 − 2 π

)

{cm}^{2}

.
Therefore, the correct answer is option (1).

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