Pappus’s centroid theorem

Theorem 1.

The surface of revolutionMathworldPlanetmath generated by a smooth curve γ in the xz-plane (with x0), rotated about the z axis, has surface areaMathworldPlanetmath


where s is the arc lengthMathworldPlanetmath of γ, and d is the distance travelled by the centroidPlanetmathPlanetmath μ of γ under a full rotation. (The centroid is also called the centre of mass, assuming the curve has uniform line density.)

Theorem 2.

The solid of revolutionMathworldPlanetmath generated by a region (Lebesgue-measurable set) S in the xz-plane (with x0) rotated about the z axis, has volume


where A is the area of S, and d is the distance travelled by the centroid μ of S under a full rotation.

In English-speaking countries, these two theorems are known as Pappus’s theorems, after the ancient Greek geometer Pappus of Alexandria. In continental Europe, these theorems are more commonly associated with the name of Paul Guldin (who rediscovered them): e.g. in German “die guldinsche Regeln”; in Finnish “Guldinin säännöt”; in French “le théorème de Guldin”.

Example 1.

The surface area of the torus, with the generating circle having radius r, and ring “radius” R (measured from the centre of the torus to the centre of the generating circle), is A=(2πr)(2πR)=4π2rR. We used here the obvious fact that the centroid of a circle is at its centre.

Example 2.

The volume of the (solid) torus, with the same parameters as above, is V=(πr2)(2πR)=2π2r2R.

Example 3.

We compute the volume of the three-dimensional ball in R3. The ball can be considered to be the solid of revolution generated by a half-disk. So we will need to know the centroid of the upper-half disk B+2(r), radius r, in the plane. By symmetryMathworldPlanetmath, this centroid has only a vertical component and no horizontal component. The vertical component is calculated by:

1πr2/2B+2(r)y𝑑x𝑑y =rπ/2B+2(1)y𝑑x𝑑y

Then the volume of the ball of radius r is

Example 4.

Similarly we can compute the surface area of the sphere of radius r, generated by revolving a half-circular arc. The centroid of the upper half-circle S+1(r) in the plane only has the vertical component:


Thus the surface area of the sphere is given by

Title Pappus’s centroid theorem
Canonical name PappussCentroidTheorem
Date of creation 2013-03-22 15:28:06
Last modified on 2013-03-22 15:28:06
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 7
Author stevecheng (10074)
Entry type Theorem
Classification msc 53A05
Synonym Guldin’s rule
Synonym Guldinus theorem
Synonym Guldin’s theorem
Synonym Pappus’s theorem for solids of revolution
Synonym Pappus’s theorem for surfaces of revolution
Related topic Centroid2
Related topic CentreOfMass
Related topic SurfaceOfRevolution2
Related topic VolumeOfSolidOfRevolution