How much is the size of space between all points with distance R to a Central point in n-dimensional space?

Answer 1

The volume of n-space enclosed by a hypersphere of radius R is:

V(n) = π^(n/2)R^n / Γ(n/2 + 1)

Where Γ represents the gamma function.

Γ(n/2 + 1) = (n/2)! for even n
Γ(n/2 + 1) = (√π)(n!!)/2^[(n+1)/2] for odd n

Where double factorial !! is:

n.(n-2) ... 5.3.1 for odd n
n.(n-2) ... 6.4.2 for even n

This link discusses n-volume of a hypersphere.

http://en.wikipedia.org/wiki/Hypersphere

You might want to consider posting this type of question under mathematics in the future. It will get a quicker response.