Problem 478

Suppose that the radius of each of the circles is $r$. Since the two circles are identical, then the two circles have equal area. Since the shaded area is common to the two circles, then the unshaded pieces of each circle have equal areas. Since the combined area of the unshaded regions equals that of the shaded region, or $216\pi$, then each of the unshaded regions have area $\frac{1}{2} \times 216\pi=108\pi$. The total area of one of the circles equals the sum of the areas of the shaded region and one unshaded region, or $216\pi+108\pi=324\pi$. Since the radius of the circle is $r$, then $\pi r^{2}=324\pi$ or $r^{2}=324$. Since $r>0$, then $r=\sqrt{324}=18$. Therefore, the circumference of each circle is $2\pi r=2\pi(18)=36\pi$.