TITLE: Calculate Volume of Torus Given Circumferences
QUESTION [6 upvotes]: What, if any, is the formula to calculate the volume of a torus given the circumference of the tube and the outer circumference of the ring?

REPLY [6 votes]: There are already two answers here, but I went ahead and computed the volume using the parametric equation of the torus, given by
\begin{align}&x=(a+r\cos v)\cos u\\&y=(a+r\cos v)\sin u\\&z= r\sin v\end{align}
where $a$ is the distance from the origin to the center of the 'tube', $r$ is the radius of the 'tube' itself, and $u$ and $v$ are two parameters corresponding to the central angle and the circular angle inside the 'tube'.

The laboriously calculated Jacobian (giving the magnitude of the volume element at every point inside the torus) is simply $ar + r^2\cos v\,dr\,dv\,du$, so the volume of the torus is given by the integral
$$\int_0^{2\pi}\int_0^{2\pi}\int_0^r ar + r^2\cos v\,dr\,du\,dv=2\pi^2r^2R$$
as desired.
Edit: Since I'm doubly bored, I went ahead and found the volume using a solid of revolution method. The torus can be considered the solid constructed by rotating a circle around the $z$ axis. The volume is given by the double integral
$$2\pi\int_{-r}^r\int_{-\sqrt{r^2-x^2}+R}^{\sqrt{r^2-x^2}+R}y\ dy\,dx=2\pi^2 r^2 R$$
Technically, the variables should be reversed, but the answer is the same.