A couple of mathematical properties of the hypersphere:

It has a hypervolume (measured in distance^4) of π^{2}r^{4}/2, and a surface volume of 2π^{2}r^{3}.

A solid angle of a hypersphere would be measured in hypersteradians, of which the hypersphere would contain 2π^{2} in total. (It would seem that 2π radians in a circle and 4π steradians in a sphere were setting up a nice simple pattern and that 8π hypersteradians would be next, but actually the N-volume, N-area, and number of N-radians of an N-sphere are all related to the Gamma function and the way it can cancel out powers of π halfway between integers.)