n-Dimensional Sphere Volume

n=1: V=2r (diameter), S=2
n=2: V=πr^2 (area of a circle), S=2πr (circumference)
n=3: V=4/3*πr^3 (volume of a sphere), S=4πr^2 (surface area of a sphere)
Calculations of the volume and surface area of spheres in four or more dimensions are used in statistical mechanics and quantum mechanics, among other fields.

\(\normalsize Sphere\ in\ n-dimensional\ space\\ (1)\ volume:\hspace{50px} V={\large\frac{\pi^{\normalsize\frac{n}{2}}r^n}{\Gamma({\large\frac{n}{2}}+1)}}\\ (2)\ surface\ area:\hspace{10px} S={\large\frac{dV}{dr}}={\large\frac{2\pi^{\normalsize\frac{n}{2}}r^{n-1}}{\Gamma({\large\frac{n}{2}})}}\\ \)