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Two circles with radius 6 pass through each other's centers. What is the area of the white region?

 

 Aug 6, 2020
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I will try to find the white area by adding together the areas of the circles and subtracting out the area of the overlap.

 

The area of each circle is  pi·62  =  36pi; so the area of the two circles is  72pi.

 

Now, for the overlap:

 

Call the center of the left-hand circle L, the center of the right-hand circle R, the top point of intersection T, and the 

bottom point of intersection B.

 

Since LT = 6, RT = 6, and LR = 6, angle(LRT) = 60o and angle(BRT) = 120o.

 

To find the segment bounded by arc(TLB) and segment(TB):

area  =  sector(TRB) - triangle(TRB)  =  (120o/360o)·36pi - ½·6·6·sin(120o)

                                                            =           12pi           -    9·sqrt(3)

 

area of both segments  =  24pi - 18sqrt(3)

 

Total white area  =  72pi - ( 24pi - 18sqrt(3) )  =  49pi + 18sqrt(3)

 Aug 6, 2020

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