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The following is problem 8 from a GRE exam found here.

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The problem states that the two circles with radius $r=3$ intersect each other such that the area of the darkened region is equal to the sum of areas of the dashed regions. Find the area of the darkened region. Thanks.

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What do you mean when you say the "area of darked figure is equal 1 sum of areas of dashed circles?" Are you saying the area of the darkened figure is equal to the sum of the areas of both dashed figures? – yunone May 31 '11 at 11:51
yes exactly it is so – dato datuashvili May 31 '11 at 11:53
Hint: that the problem uses circles isn't really important, it would work just as well with two overlapping squares or rectangles or triangles (with the same area). – yatima2975 May 31 '11 at 11:56
I hope you don't mind if I tried to improve the formatting of your question user3196. If anything is not to your liking, please don't hesitate to rollback. – yunone May 31 '11 at 12:10
up vote 6 down vote accepted

Denote the area of the darkened region as $A$, and denote the area of each of the dashed regions as $B$. The areas of the dashed regions are equal since the two intersecting circles have equal area. So $A+B=\pi\cdot 3^2=9\pi$, since $A$ and $B$ added give the area of the circle with radius $3$. But based on given information, $A=2B$. Can you proceed from there?

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yes it means that area of darkened circle is 6*PI which is also in answer too thanks once again.i am preparing for exams in general abilities for gain grant from states and it is exams for master programs thanks @yunone – dato datuashvili May 31 '11 at 12:00
@user3196 glad to help. Good luck with your preparation. – yunone May 31 '11 at 12:02

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