The Second Theorem of Pappus is in the same spirit as Pappus’s Theorem discussed in this section, but for surface area rather than volume: let C be a curve that lies entirely on one side of a line l in the plane. If C is rotated about l, then the area of the resulting surface is the product of the arc length of C and the distance traveled by the centroid of C.

Use the Second Theorem of Pappus to find the surface area of the torus in Example 7.

EXAMPLE 7

A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R(> r) from the center of the circle. Find the volume of the torus.

The circle has area A = πr². By the symmetry principle, its centroid is its center and so the distance traveled by the centroid during a rotation is d = 2πR. Therefore, by the Theorem of Pappus, the volume of the torus is

V = Ad = (2πR)(πr²) = 2π²r²R