I'm glad to hear it! I might be biased, but I do think it's pretty cool that you're able to achieve such a result. Also—while this may be a bit trivializing in some sense, we really do have a function for everything! Try out RegionMeasure, if you'd like, e.g.:
Manipulate[
RegionMeasure /@ {Disk[{0, 0}, d], Sphere[{0, 0, 0}, d],
Ball[{0, 0, 0}, d], Cuboid[{0, 0, 0}, d {1, 1, 1}]}, {d, 1, 5, 1}]
This can be useful to provide some insight which is relevant to the dimensionality question we're discussing (somewhat paradoxically). On a related note, you might try something like Volume[Sphere[]] and get a (previously-)surprising result.
Or, you could make the variation continuous, then Plot to see the problem another way:
Plot[Evaluate[
RegionMeasure /@ {Disk[{0, 0}, d], Sphere[{0, 0, 0}, d],
Ball[{0, 0, 0}, d], Cuboid[{0, 0, 0}, d {1, 1, 1}]}], {d, 1, 5},
ScalingFunctions -> {"Log", "Log"}]
(adding PlotLegends->"Expressions" would make it explicit, to wit.)
EDIT: I'd be loathe to forget RegionDimension!
