Pappus's centroid theorem

File:Pappus centroid theorem areas.gif

The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation.

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria and Paul Guldin.

The first theorem

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid.

$A = sd.\,$

For example, the surface area of the torus with minor radius r and major radius R is

$A = (2\pi r)(2\pi R) = 4\pi^2 R r.\,$

The second theorem

The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid.

$V = Ad.\,$

For example, the volume of the torus with minor radius r and major radius R is

$V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2.\,$

Generalizations

The theorem can be generalized for arbitrary curves and shapes, under appropriate conditions.

External links

• Weisstein, Eric W., "Pappus's Centroid Theorem", MathWorld.