Comment by evanb
This is a nice analog but unfortunately I think it breaks down in a way the "π" calculation does not.
In the article 2π(d) = the ratio of the circumference to the radius. This is dimensionless, in the sense that the circumference and the radius are both lengths (measured in meters, or whatever), so 2π(d) is really just a number.
But the (hyper)volumes you're talking about depend on dimension, which is exactly why you say "hyper". In 2 dimensions the volume is the area, πr^2, which has dimensions L^2 [measured in m^2 or whatever]. But in 3 dimensions the volume is 4/3 πr^3, which has dimensions L^3. The 5 dimensional (hyper)volume has dimensions L^5, and so on.
So, "comparing" these to find out which is bigger and which smaller is not really meaningful---just like you shouldn't ask which is the bigger mass: a meter or a second? Neither is, they aren't masses.
Yeah, though they are somewhat similar. We could perhaps say the volume of the unit ball is (in some sense) "larger" than the area of the unit disk, because both are measured relative to a radius of 1. So the area of the disk is fewer "units" than the volume of the ball.
Anyway, it seems independently interesting that this value peaks for the 5-ball, or the ~5.2569-ball. The non-fractional difference between fractional dimension of peak hyper volume and peak hyper surface area seems also interesting. (I assume there is some trivial explanation for this though.)