The volume of a unit n-sphere is
and the volume of an n-cube width half-side length 1 is
Here is a table of the ratio of volume of the cube to that of the sphere (for sphere calculations, see hypersphere Eric Weisstein's World of Math).
Therefore, it can be seen that discarding factors of order this ratio becomes unacceptable for dimensions higher than
References
Barenblatt, G. I. Dimensional Analysis. New York: Gordon and Breach Science Publishers, 1987.
Barenblatt, G. I. Scaling: Dimensional Analysis and Intermediate Asymptotics. 1996.
Bridgman, P. W. Dimensional Analysis. New Haven, CT: Yale University Press, 1963.
Weisstein, E. W. "Books about Dimensional Analysis." http://www.ericweisstein.com/encyclopedias/books/DimensionalAnalysis.html.
