Torus Area and Volume

A torus or anchor ring, drawn in Fig. 1.3, is the approximate shape of a donut or bagel. The radii R and r refer, respectively, to the circle through the center of the torus and the circle made by a cross-sectional cut. Generally, to determine the area and volume of a surface of revolution, it is necessary to evaluate double or triple integrals. However, long before calculus was invented. Pappus of Alexandria (ca. Third Century A.D.) proposed two theorems that can give the same results much more directly. 

Analogously, the second theorem of Pappus states that the volume of the solid generated by the revolution of a figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. For a torus, the area of the cross section equals nr. Therefore, the volume of a torus is given by 

For less symmetrical figures, finding the centroid will usually require doing an integration over the cross section. 

The surface of revolution with radius Rofa circle of radius r. 

An incidental factoid. You probably know about the four-color theorem on a plane or spherical surface, four colors suffice to draw a map in such a way that regions sharing a common boundary have different colors. On the surface of a torus it takes seven colors. 

---