Volume and surface area of torus.
A torus has the shape of a doughnut. A torus is a surface of revolution generated by revolving a circle in three-dimensional space about a line that does not intersect the circle.
Torus surface area and volume are calculated by the Pappus's centroid theorems:
, where s is for the arc length of the curve being rotated, d is for the distance traveled by the centroid of the region in one rotation,
, where A is for area of the region being rotated, d is for the distance traveled by the centroid of the region in one rotation,
So surface area of torus:
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 is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus.