 TOPICS # 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.

## Referenced on Wolfram|Alpha

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