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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.

solidgenerating curvesx^_S
coneinclined line segmentsqrt(r^2+h^2)1/2rpirsqrt(r^2+h^2)
cylinderparallel line segmenthr2pirh
spheresemicirclepir(2r)/pi4pir^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.

solidgenerating laminaAx^_V
coneright triangle1/2hr1/3r1/3pir^2h
cylinderrectanglehr1/2rpir^2h
spheresemicircle1/2pir^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

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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 Resource. https://mathworld.wolfram.com/PappussCentroidTheorem.html

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