N-dimensional Circles!
Mathematica, 18 bytes, up to ~168.15 trillion dimensions

Pi^(a=.5#)/a!#2^#&amp;


Anonymous function. Takes two numbers as input, and returns an inexact number as output. Works with any number of dimensions. Outputs 1. for n = 0. Uses the formula from Volume of an n-ball on Wikipedia.

Explanation

We are attempting to compute &pi;n/2/&Gamma;(n/2 + 1)&middot;Rn, or N[Pi^(n/2)/Gamma[n/2 + 1] R^n] in Mathematica. In our case, # (first argument) is n and #2 (second argument) is R. This leaves us with N[Pi^(#/2)/Gamma[#/2 + 1] #2^#] &amp;, which can be golfed as follows:

N[Pi^(#/2)/Gamma[#/2 + 1] #2^#] &amp;
Pi^(.5#)/Gamma[.5# + 1] #2^# &amp;    (* replace exact with approximate numbers*)
Pi^(.5#)/(.5#)! #2^# &amp;            (* n! == Gamma[n + 1] *)
Pi^(a=.5#)/a! #2^# &amp;              (* replace repeated .5# *)
Pi^(a=.5#)/a!#2^#&amp;                (* remove whitespace *)


and thus, our original program.