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# In the figure above, if the square inscribed in the circle h

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In the figure above, if the square inscribed in the circle h [#permalink]  04 Jun 2016, 07:12
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#GREpracticequestion In the figure above, if the square inscribed in the circle.jpg [ 11.95 KiB | Viewed 8670 times ]

In the figure above, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?

A. $$2\pi – 1$$

B. $$2\pi – 4$$

C. $$4\pi – 2$$

D. $$4\pi – 4$$

E. $$8\pi – 4$$

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[Reveal] Spoiler: OA

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Re: In the figure above, if the square inscribed in the circle h [#permalink]  04 Jun 2016, 07:21
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Explanation

Attachment:

#GREpracticequestion In the figure above, if the square inscribed in the circle.jpg [ 16.38 KiB | Viewed 6668 times ]

If the area of the square is 16, that means each side is 4-

Now we do have an isosceles triangle with two equal side

$$x:x:x\sqrt{2}$$

$$4 : 4 : 4 \sqrt{2}$$

So the side of the triangle which is also the circumference. Therefore, the radius is half that $$\frac{4 \sqrt{2}}{2} = 2 \sqrt{2}$$ and the area of the circle is $$\pi r^2=\pi(2 \sqrt{2} )^2 = 8 \pi$$

The area we are looking for is $$\frac{8\pi - 16}{4} = 2\pi - 4$$

B is correct
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Re: In the figure above, if the square inscribed in the circle h [#permalink]  05 Aug 2019, 08:16
Can you explain further detail for the steps after finding 8π? I understand the values but I don't understand why.
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Re: In the figure above, if the square inscribed in the circle h [#permalink]  05 Aug 2019, 08:56
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ehilario wrote:
Can you explain further detail for the steps after finding 8π? I understand the values but I don't understand why.

8π is the area of the circle
Since the area of the square (that's inside the circle) is 16, then 8π - 16 represents the area of ALL 4 tiny circle pieces.
Since we just want the area of ONE tiny circle piece, we need to divide 8π - 16 by 4

Does that help?

Cheers,
Brent
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Re: In the figure above, if the square inscribed in the circle h [#permalink]  05 Aug 2019, 09:43
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Yes it does, thank you!

GreenlightTestPrep wrote:
ehilario wrote:
Can you explain further detail for the steps after finding 8π? I understand the values but I don't understand why.

8π is the area of the circle
Since the area of the square (that's inside the circle) is 16, then 8π - 16 represents the area of ALL 4 tiny circle pieces.
Since we just want the area of ONE tiny circle piece, we need to divide 8π - 16 by 4

Does that help?

Cheers,
Brent
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Re: In the figure above, if the square inscribed in the circle h [#permalink]  07 Oct 2020, 00:09
sandy wrote:
Explanation

Attachment:
#GREpracticequestion In the figure above, if the square inscribed in the circle.jpg

If the area of the square is 16, that means each side is 4-

Now we do have an isosceles triangle with two equal side

$$x:x:x\sqrt{2}$$

$$4 : 4 : 4 \sqrt{2}$$

So the side of the triangle which is also the circumference. Therefore, the radius is half that $$\frac{4 \sqrt{2}}{2} = 2 \sqrt{2}$$ and the area of the circle is $$\pi r^2=\pi(2 \sqrt{2} )^2 = 8 \pi$$

The area we are looking for is $$\frac{8\pi - 16}{4} = 2\pi - 4$$

B is correct

can you explain why you divide 8π - 16 with 4?
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Re: In the figure above, if the square inscribed in the circle h [#permalink]  07 Oct 2020, 05:24
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sleepyowl wrote:
sandy wrote:
Explanation

Attachment:
#GREpracticequestion In the figure above, if the square inscribed in the circle.jpg

If the area of the square is 16, that means each side is 4-

Now we do have an isosceles triangle with two equal side

$$x:x:x\sqrt{2}$$

$$4 : 4 : 4 \sqrt{2}$$

So the side of the triangle which is also the circumference. Therefore, the radius is half that $$\frac{4 \sqrt{2}}{2} = 2 \sqrt{2}$$ and the area of the circle is $$\pi r^2=\pi(2 \sqrt{2} )^2 = 8 \pi$$

The area we are looking for is $$\frac{8\pi - 16}{4} = 2\pi - 4$$

B is correct

can you explain why you divide 8π - 16 with 4?

The area of the circle is 8π
The area of the inscribed Square is 16
So, 8π - 16 = the area of the FOUR partial circles (one of which is shaded)
So to find the area of the ONE shaded partial circle, we must divide by 4
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Re: In the figure above, if the square inscribed in the circle h   [#permalink] 07 Oct 2020, 05:24
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