A parameterization for the torus is given by
, where .
Here R is the major radius and s is the minor radius. Basically you can construct this parameterization by beginning with the large circle .
This parameterization can be extended to the torus by adding smaller circles of radius s at each point.
By the way, you can calculate the volume of a torus simply by transforming it into a cylinder. Given a major radius R its height is then 2 pi R and the base has a surface of pi s^2 where s is the minor radius.