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Retired Moderator Joined: 29 Apr 2015
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A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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11 00:00

Difficulty:   45% (medium)

Question Stats: 71% (02:29) correct 29% (02:55) wrong based on 216 sessions

### HideShow timer Statistics Attachment: T8916.png [ 4.8 KiB | Viewed 5174 times ]

A square is inscribed within a circle, as shown above. If the total area of the shaded regions is 2, what is the area of the circle?

A. 2pi/(pi-2)
B. 2pi/p-1)
C. pi/(pi-2)
D. pi/(pi-4)
E. 2pi/(pi-4)

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Originally posted by reto on 30 May 2015, 06:22.
Last edited by Bunuel on 31 May 2015, 06:00, edited 1 time in total.
Edited the question.
Manager  Joined: 27 Dec 2013
Posts: 222
Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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HI. I am getting the answer 4PI/PI-2.

Suppose side of the square=a. then diagnal= \sqrt{2} X a.

Divide the diagnoal by 2= You will be get= \sqrt{2} Xa /2= a/\sqrt{2}

i am getting the answer 4PI/PI-2.

reto wrote:
Attachment:
T8916.png

A square is inscribed within a circle, as shown above. If the total area of the shaded regions is 2, what is the area of the circle?

A. 2pi/pi-2
B. 2pi/p-1
C. pi/pi-2
D. pi/pi-4
E. 2pi/pi-4

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Kudos to you, for helping me with some KUDOS.
Manager  Joined: 17 Mar 2015
Posts: 116
Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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3
Lets assume side of the square = $$a$$, diagonal is $$a*\sqrt{2}$$ so radius is half of it - $$\frac{a}{2}*\sqrt{2}$$
$$2 = \pi*\frac{a^2}{2} - a^2$$ (1)
What we need to find is: S = $$\pi*\frac{a^2}{2}$$
From the (1): $$a^2 = 2/(\pi/2 - 1)$$
put it in our resulting equation for the answr:
$$S = \pi/2*2/(\pi/2 - 1)= 2*\pi/(\pi - 2)$$ which corresponds to A
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Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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Another way to solve such question with shaded regions is to ballpark the answer choices. In the GMAT questions with shaded areas are there because these areas has no names.

Start with a snapshot ballpark. If the shaded region is 2, estimate how big the whole circle must be. In this example it's about 6 from guessing.

Generally for shaded regions in the GMAT:

Shaded regions - Subtraction with pi
• In the GMAT, an area is shaded when there is no geometric term to describe it directly.
• In such cases that area can be calculated as a subtraction of one shape from another.
• In the case there is a circular rim and a straight edge - The GMAT uses a subtraction involving a circle and a non-circular shape.
• Since does not cancel out, whenever an exact result is required you can:
o And of course- as always, POE answers out of the ballpark
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Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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Hi shriramvelamuri,

Did the above explanations clear this up for you? I was trying to see where your approach broke down, but I don't see how you got from diagonal = a/2rt.2 to your answer.
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Posts: 56277
Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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reto wrote:
Attachment:
T8916.png

A square is inscribed within a circle, as shown above. If the total area of the shaded regions is 2, what is the area of the circle?

A. 2pi/pi-2
B. 2pi/p-1
C. pi/pi-2
D. pi/pi-4
E. 2pi/pi-4

Check other Shaded Region Problems in our Special Questions Directory.
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A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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Hey Bunuel,

Came across this question today and could not figure it out. Can you break it down for us? Appreciated.
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Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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1
Hey Bunuel,

Came across this question today and could not figure it out. Can you break it down for us? Appreciated.

Let r be the radius of the circle and 'a' be the side of the square.

As shown in the attached figure, in the right triangle drawn:

$$r^2 = (a/2)^2 + (a/2)^2$$ ---> $$a^2 = 2r^2$$

From the given figure,

Shaded area = area of the circle - area of the square ---> $$\pi * r^2 - a^2 = 2$$ ----> $$\pi * r^2 - 2r^2 = 2$$ ----> $$r^2 = \frac{2}{\pi -2}$$

Thus, area of the circle = $$\pi*r^2 = \frac{2\pi}{\pi -2}$$ . A is the correct answer.
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Posts: 2
Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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Engr2012 wrote:
Hey Bunuel,

Came across this question today and could not figure it out. Can you break it down for us? Appreciated.

Let r be the radius of the circle and 'a' be the side of the square.

As shown in the attached figure, in the right triangle drawn:

$$r^2 = (a/2)^2 + (a/2)^2$$ ---> $$a^2 = 2r^2$$

From the given figure,

Shaded area = area of the circle - area of the square ---> $$\pi * r^2 - a^2 = 2$$ ----> $$\pi * r^2 - 2r^2 = 2$$ ----> $$r^2 = \frac{2}{\pi -2}$$

Thus, area of the circle = $$\pi*r^2 = \frac{2\pi}{\pi -2}$$ . A is the correct answer.

Thank you! Super clear now.
Intern  Joined: 28 Jul 2015
Posts: 3
A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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1
Area of the shaded region is equal to: $$∏r^2 (circle)-a^2 (square)$$
Since we know that 2r is diagonal of the square, we can get the side of the square by using Pythagorem Theorem:
$$a^2+a^2=〖(2r)〗^2$$
$$2a^2 = 4r^2$$
$$a^2 = 2r^2$$
$$a=√2rr$$
$$a=r*√2$$

The shaded area equals to 2. So:

$$∏r^2-a^2=2$$
$$∏r^2- (r*√2)^2=2$$
$$∏r^2-r^2*2=2$$
$$r^2*(∏-2)=2$$
$$r^2=\frac{2}{(∏-2)}$$

So the area of the circle is equal to $$r^2*∏$$
That gives us area of the circle = $$\frac{2}{(∏-2)} *∏$$

So area of the circle is $$\frac{(2∏)}{((∏-2))}$$
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Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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If r is the radius of a circle and a square is inscribed in it, the area of the square is 2r^2. Why? Because r is the radius and there can be 4 triangles formed inside such a square when diagonals (diameters) cross each other. Each triangle has an area = 1/2 r*r = > 1/2r^2. So the square's area = 2r^2.

area of circle - area of square = 2
if we re-arrange r^2 to one side of the equation, we will get choice A.
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Joined: 19 Jul 2017
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Re: A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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reto wrote:
Attachment:
T8916.png

A square is inscribed within a circle, as shown above. If the total area of the shaded regions is 2, what is the area of the circle?

A. 2pi/(pi-2)
B. 2pi/p-1)
C. pi/(pi-2)
D. pi/(pi-4)
E. 2pi/(pi-4)

A general equation for the area of the shaded region
= πr^2 - 2r^2, where r = radius of the circle.

So, πr^2 - 2r^2=2
r^2=2/pi-2
πr^2=2pi/(pi-2)
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A square is inscribed within a circle, as shown above. If the total ar  [#permalink]

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1
reto wrote:
Attachment:
T8916.png

A square is inscribed within a circle, as shown above. If the total area of the shaded regions is 2, what is the area of the circle?

A. 2pi/(pi-2)
B. 2pi/p-1)
C. pi/(pi-2)
D. pi/(pi-4)
E. 2pi/(pi-4)

can also be calculated as

The area of the shaded region/4 = area of sector - area of triangle

since we have a square and each of the shaded region has same area
we pick one of the shaded region
area of one such shaded region = 2/4
also,
let a be the side of the square
the diagonal therefore will be a*root2 = diameter of the circle
so
2/4=pi*r^2*90/360 - 1/2(a*root2/2)*(a*root2/2)
2/4=p1*2a^2/16 - a^2/4
this gives
a=4/2pi-1
now area of the circle = pi*r^2
and r = a*root2/2
substitute for r
we get 2pi/2pi-1 A square is inscribed within a circle, as shown above. If the total ar   [#permalink] 25 Jul 2018, 12:00
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