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Pappus's Centroid Theorem

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.

 solid generating curve cone inclined line segment cylinder parallel line segment sphere semicircle

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.

 solid generating lamina cone right triangle cylinder rectangle sphere semicircle

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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Pappus's Centroid Theorem

Cite this as:

Weisstein, Eric W. "Pappus's Centroid Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PappussCentroidTheorem.html