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What is the approximate area of the shaded region?

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by swerve » Wed Nov 22, 2017 5:40 pm
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Square ABCD is perfectly inscribed in the circle pictured above. If minor arc AD measure 2Ï€, what is the approximate area of the shaded region?

A. 110
B. 72
C. 36
D. 18
E. 9

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.
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by Jay@ManhattanReview » Wed Nov 22, 2017 8:14 pm
swerve wrote:Image

Square ABCD is perfectly inscribed in the circle pictured above. If minor arc AD measure 2Ï€, what is the approximate area of the shaded region?

A. 110
B. 72
C. 36
D. 18
E. 9

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.
The area of the shaded region = Area of the circle - Area of the square

Since the square is perfectly inscribed, Arc AD = Arc DC = Arc CB = Arc BA

Thus, length of arc AD = 1/4 of circumference of the circle

=> 2Ï€ = (2Ï€r)/4 => r = 4

=> Area of the circle = πr^2 = π.4^2 = 16π

Side of the square would be given by √2r

Area of the square = (Side)^2 = (√2r)^2 = 2r^2 = 2.4^2 = 32

Thus, the area of the shaded region = Area of the circle - Area of the square = 16Ï€ - 32 = 16.(3.14) - 32 = ~50 - 32 = ~18.

The correct answer: D

Hope this helps!

-Jay
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by Scott@TargetTestPrep » Sat Oct 12, 2019 4:02 pm
swerve wrote:Image

Square ABCD is perfectly inscribed in the circle pictured above. If minor arc AD measure 2Ï€, what is the approximate area of the shaded region?

A. 110
B. 72
C. 36
D. 18
E. 9

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.
Since minor arc AD is 1/4 of the circle, the circumference of the circle is 4 x 2Ï€ = 8Ï€.

Now we can find the radius of the circle using the circumference formula C = 2Ï€r:

8Ï€ = 2Ï€r

4 = r

Thus, the area of the circle is π * 4^2 = 16π and the diameter of circle = 8. Since the diameter of circle = the diagonal of the square and the diagonal of the square = side√2, we have:

8 = side√2

8/√2 = side

Thus, the area of the square is (8/√2)^2 = 64/2 = 32.

So, the area of the shaded region is approximately 16π - 32 = 16(π - 2). Since π is approximately 3.14, we have 16(3.14 - 2) = 16(1.14), which is approximately 18.

Answer: D

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