A torus has the shape of a doughnut. Formally, a torus is a surface of revolution generated by revolving a circle in three dimensional space about a line which does not intersect the circle.

Torus surface area and volume are calculated by the Pappus's centroid theorems:

, where s - the arc length of the curve being rotated, d is the distance traveled by the centroid of the region in one rotation,
, where A - area of the region being rotated, d is the distance traveled by the centroid of the region in one rotation,


So surface area of torus:
Torus volume:


The three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the standard ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus.