The ring option is fairly easy to calculate: each of the nine circles will have an area of pi/4, therefore the shaded are is 9pi/4 = 7.069…

The crescent area is trickier.

The radius of the largest shaded circle is 1, the next is 2/3, the next 1/3, then 2/11… Naming the largest circle the zeroth, in general the radius of the nth circle is:

2/(n^2 + 2). The area of each circle is pi.r^2 or:

pi*(2/(n^2+2))^2. Usefully this also works for negative values of n, so will cover both arms of the crescent.

At this point I consulted the Wolfram Alpha website to see if the infinite sum of this boils down to anything simple:

sum_(n=-∞)^∞ π (2/(n^2 + 2))^2 = 1/2 π^2 (sqrt(2) coth(sqrt(2) π) + 2 π csch^2(sqrt(2) π))

Either using Wolfram Alpha or Excel we can estimate the infinite sum as ≈ 6.998…

Therefore the ring option gives a slightly larger shaded area.