Pappus's Centroid Theorem

Pappus's centroid theorem relates the surface area or volume of a figure generated by rotation to the geometric centroid of its generating curve or lamina. The first theorem states that the surface area of a surface of revolution generated by the rotation 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 ,

(1)

(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 rotation 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 ,

(2)

(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 of revolution and surfaces of revolution.

solid	generating lamina
cone	right triangle
cylinder	rectangle
sphere	semicircle

The geometric centroid of the generated solid of revolution can also be obtained from the generating lamina (Cloete 2023, p. 322). Let a bounded lamina in the -half-plane be rotated about the z-axis through , with measured from the positive x-axis. Write , , and . The volume is , and the geometric centroid has Cartesian coordinates

			(3)
			(4)
			(5)

For a surface of revolution generated by a simple piecewise smooth curve of finite arc length, the same formulas hold after replacing by the arc length element in the three integrals (Cloete 2023, p. 324). In this case, is the surface area of the swept surface, excluding any end faces. Both versions assume uniform density and no repeated covering.