For the surface area and volume of a unit n-dimensional hypersphere, we have the formulas:

S=τ^floor(n/2)/(n-2)!! ×2 if n is odd

V=τ^floor(n/2)/n!! ×2 if n is odd

These are simple and easy to calculate and don't require a gamma function.

Additionally, tau simplifies the formulas for hypertori as well, and I'm surprised no one ever seems to talk about them. For an n-torus (here, n is the dimension of the surface, not the interior), the formulas are

S=τ^n times all the radii

V=1/2*τ^n times all the radii (with the last one squared)

These are extremely simple in terms of tau!