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A **surface of revolution** is formed by the rotation of a planar
curve *C* about an axis in the plane of the curve and not cutting the
curve. The **Pappus--Guldinus theorem** says that:

- The
**area of the surface of revolution**on a curve*C*is equal to the product of the length of*C*and the length of the path traced by the centroid of*C*(which is 2 the distance from this centroid to the axis of revolution). - The
**volume bounded by the surface of revolution**on a simple closed curve*C*is equal to the product of the area bounded by*C*and the length of the path traced by the centroid of the area bounded by*C*.

When *C* is a circle, the surface obtained is a **circular torus** or
**torus of revolution** (Figure 1). Let *r* be the
radius of the revolving circle and let *R* be the distance from its
center to the axis of rotation. The **area** of the torus is
4*Rr*, and its **volume** is 2*Rr*.

**Figure 1:** A torus of revolution.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.