Volume of Torus

Theorem

Formulation 1

Let $\TT$ be a torus.

Let $r$ be the radius of the generating circle of $\TT$.

Let $R$ be the distance of the center of the generating circle from the axis of revolution of $\TT$.

Then the volume $\VV$ enclosed by $\TT$ is given by:

$\VV = 2 \pi^2 r^2 R$

Formulation 2

Let $\TT$ be a torus.

Let $a$ and $b$ be the inner radius and outer radius respectively of $\TT$.

Then the volume $\VV$ enclosed by $\TT$ is given by:

$\VV = \dfrac {\pi^2 \paren {a + b} \paren {b - a}^2} 4$