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 |