Pappus's Centroid Theorem

The Surface Area of a Surface of Revolution is given by

$$S_{\rm solid\ of\ rotation} = \hbox{[perimenter]}\times\hbox{[distance traveled by centroid]},$$

and the Volume of a Solid of Revolution is given by

$$V_{\rm solid\ of\ rotation} = \hbox{[cross-section area]}\times \hbox{[distance traveled by centroid]}.$$

See also Centroid (Geometric), Cross-Section, Perimeter, Solid of Revolution, Surface Area, Surface of Revolution, Toroid, Torus

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987.