Shady Business

The area of the larger circle is $9\pi$ .

The area of the smallest circle is $\pi$ .

The area of the other two circles is $4\pi +4\pi$ .

Thus, the area of the small and medium circles combined is $\pi +4\pi+4\pi=9\pi$ .

Let $x$ be the area of the white part; thus, the area of the horizontally shaded area ( $B)$ is ( $9\pi-x$ ),

and the area of the vertically shaded area ( $A)$ is $9\pi-x$ .

We can conclude that $A = B$ .

Let $W$ be the combined area of the three unshaded regions. The area of the large circle and the combined areas of the smaller circles are both $36\pi$ , so $A = B = 36\pi - W$ .

As the amount of overlap is not specified, it doesn’t change the result. Therefore assume no overlap (circles are externally tangent, or disjoint). Then white area = 0, area of large circle = 9 pi, area of middle circles = 4 pi, area of small circle = pi. Sum of areas of small circle and 2 middle circles = pi + 4 pi + 4 pi = 9 pi. This equals area of large circle, 9 pi. Thus red and blue areas are equal.