The first theorem of Pappus states that the surface area 
of a surface of revolution generated by the
revolution of a curve about an external axis is equal to the product of the arc
length 
of the generating curve and the distance 
traveled by the curve's geometric
centroid 
,
(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.
Similarly, the second theorem of Pappus states that the volume 
of a solid of revolution generated by the revolution
of a lamina about an external axis is equal to the product of the area 
of the lamina and the distance 
traveled by the lamina's geometric
centroid 
,
(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem for various solids and surfaces of revolution.
See also
Cross Section, Geometric Centroid, Pappus Chain, Pappus's Harmonic Theorem, Pappus's Hexagon Theorem, Perimeter, Solid of Revolution, Surface Area, Surface of Revolution, Toroid, Torus
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References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987.Harris, J. W. and Stocker, H. "Guldin's Rules." §4.1.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 96, 1998.Kern, W. F. and Bland, J. R. "Theorem of Pappus." §40 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 110-115, 1948.
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Cite this as:
Weisstein, Eric W. "Pappus's Centroid Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PappussCentroidTheorem.html