Pappus's Centroid Theorem
PappussCentroidTheorem

The first theorem of Pappus states that the surface area S 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 s of the generating curve and the distance d_1 traveled by the curve's geometric centroid x^_,

 S=sd_1=2pisx^_

(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 s x^_ S
cone inclined line segment sqrt(r^2+h^2) 1/2r pirsqrt(r^2+h^2)
cylinder parallel line segment h r 2pirh
sphere semicircle pir (2r)/pi 4pir^2
PappussCentroidTheorem2

Similarly, the second theorem of Pappus states that the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d_2 traveled by the lamina's geometric centroid x^_,

 V=Ad_2=2piAx^_

(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 A x^_ V
cone right triangle 1/2hr 1/3r 1/3pir^2h
cylinder rectangle hr 1/2r pir^2h
sphere semicircle 1/2pir^2 (4r)/(3pi) 4/3pir^3

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

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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