I don't think this is right. If you're worried about units you can calculate the (generalized) surface area to volume ratio, which turns out to be exactly D/r. In other words, as D increases, the ratio goes to infinity.

I think this fact can fairly be interpreted to mean that a high-dimensional unit sphere encloses almost no volume. The 2D cartoon drawing of a hypersphere also helps capture this: you can imagine the "spikes" stretching out and squeezing the interior portion, until it's all outside and no inside.

EDIT: another argument I've seen involves calculating the ratio of the volume of a thin shell surrounding the n-sphere's surface to its total volume. You can prove that the limit of the ratio as the dimension goes to infinity is 1. In other words, in high dimensions almost all of the volume of the sphere is concentrated near its surface.