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