Volume of spheres in higher dimensions? What is the volume of spheres in higher dimensions?
Did you try the obvious?
In the real world, spheres are measured by their diameter, not their radius.  Were one to buy tyres, or plates, or ironmongery (bolts, nuts, drills, that sort of thing), and holes of all sorts, are measured by their diameter.

Moreover, there are named units corresponding to the area of an inch (diameter) circle, and the volume of an inch diameter sphere - circular inch, and spheric inch.  This is because the three regular solids that are the Nth power of a line, all define valid products, and coherent units.

Thus, we have the conversion factor in rexx style.

   C2P(N) = PPI( 1/FACT(N, 2),  N // 2)
       PPI(a, b) = a * (PI/2)^2
       FACT(a, b) =  a * (a-b) * (a-2b) ... while a-nb &gt;0


In mathematical terms, one could write $c2p(n) = (\pi/2)^{n//2} / (n-1)!!$, and the volume of a sphere is then $vol(sph) = 2^N * c2p(n)$