What is the approximate area of the shaded region?

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swerve: Legendary Member; Posts: 2535; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Wed Nov 22, 2017 5:40 pm

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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Source: Beat The GMAT — Problem Solving | Wed Nov 22, 2017 5:40 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Wed Nov 22, 2017 8:14 pm

swerve wrote:

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 = Ï€r^2 = Ï€.4^2 = 16Ï€

Side of the square would be given by âˆš2r

Area of the square = (Side)^2 = (âˆš2r)^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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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sat Oct 12, 2019 4:02 pm

swerve wrote:

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 Ï€ * 4^2 = 16Ï€ and the diameter of circle = 8. Since the diameter of circle = the diagonal of the square and the diagonal of the square = sideâˆš2, we have:

8 = sideâˆš2

8/âˆš2 = side

Thus, the area of the square is (8/âˆš2)^2 = 64/2 = 32.

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

Answer: D

Scott Woodbury-Stewart
Founder and CEO
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