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Three congruent circles overlap in such a way that each circ

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Three congruent circles overlap in such a way that each circ [#permalink]  11 Mar 2013, 23:48
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Three congruent circles overlap in such a way that each circle intersects the centers of both of the other circles, as shown below. If the radius of each of the circles is 8, what is the area of the central section where all three circles overlap?

A. 16\sqrt{3}

B. 32(\pi-\sqrt{3})

C. 16(\pi+\sqrt{3})

D. 32\pi

E. 32(\pi+\sqrt{3})
[Reveal] Spoiler: OA

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Last edited by Bunuel on 12 Mar 2013, 01:53, edited 1 time in total.
Edited the question.
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Re: Three congruent circles overlap in such a way that each circ [#permalink]  12 Mar 2013, 02:10
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emmak wrote:
Attachment:
The attachment 2.jpg is no longer available
Three congruent circles overlap in such a way that each circle intersects the centers of both of the other circles, as shown below. If the radius of each of the circles is 8, what is the area of the central section where all three circles overlap?

A. 16\sqrt{3}

B. 32(\pi-\sqrt{3})

C. 16(\pi+\sqrt{3})

D. 32\pi

E. 32(\pi+\sqrt{3})

I'd go with approximation with this question. Look at the diagram below:
Attachment:

Circles.png [ 21.94 KiB | Viewed 2395 times ]
Notice that triangle ABC is equilateral (all sides are radii of the circles, for example AB and AC are radii of the lower circle and BC is radius of both upper circles). This implies that angle A is 60 degrees, thus the area of sector ABC (yellow region in the lower circle) is 1/6th of the area of the circle, so it's area is \frac{8^2\pi}{6}\approx{34}.

Now, the area of the region we need to find must be more than this but not too much. Only B fits.

Hope it's clear.
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Re: Three congruent circles overlap in such a way that each circ [#permalink]  13 Mar 2013, 19:19
The area of the overlapping section should be little less than the half of the area of the circle , and only B fits
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Re: Three congruent circles overlap in such a way that each circ [#permalink]  13 Mar 2013, 19:43
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Expert's post
emmak wrote:
Attachment:
2.jpg
Three congruent circles overlap in such a way that each circle intersects the centers of both of the other circles, as shown below. If the radius of each of the circles is 8, what is the area of the central section where all three circles overlap?

A. 16\sqrt{3}

B. 32(\pi-\sqrt{3})

C. 16(\pi+\sqrt{3})

D. 32\pi

E. 32(\pi+\sqrt{3})

To get the exact answer, we can just follow Bunuel's solution a little further.

Area of overlap = Area of sector ABC + 2 (Area of sector ABC - Area of triangle ABC) (or you can also look at it as Area of overlap = 3*Area of sector ABC - 2*Area of triangle ABC)

Area of overlap = \frac{64\pi}{6} + 2(\frac{64\pi}{6} - \frac{\sqrt{3}*64}{4})
Area of overlap = 32(\pi - \sqrt{3})
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Re: Three congruent circles overlap in such a way that each circ [#permalink]  24 Jun 2014, 11:22
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Re: Three congruent circles overlap in such a way that each circ   [#permalink] 24 Jun 2014, 11:22
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