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Theorem of Pappus

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The theorem of Pappus can be either one of two related theorems that can help us derive formulas for the volumes and surface areas of solids or surfaces of revolution.

They are named after Pappus of Alexandria, who worked on them.

The First Theorem of Pappus

The first theorem of Pappus tells us about the surface area of the surface of revolution we get when we rotate a plane curve around an axis which is external to it but on the same plane. It tells us that the surface area (A) of this surface of revolution is equal to the product of the arc length of the generating curve (s) and the distance d traveled by the curve’s geometric centroid. That is:

A = s * d

A torus (donut shape) with a minor radius of r and a major radius of R will have a generating curve of 2πr and the distance traveled by the curve’s geometric centroid will be 2πR. So the surface area will be

A = (2 π r)(2 π R) = 4 π2 r R

The animation below shows how this theorem applies to three surfaces of revolution: an open cylinder, a cone, and a sphere. Here, the centroids are shown by the dots, and are a distance a (shown in red) from the axis of rotation.
theorem of pappus three surfaces

The Second Theorem of Pappus

The second theorem of Pappus is very similar to the first; It tells us that the volume of a solid of revolution which is generated by rotating a plane figure about an external axis equals the area of the plane figure (call this A) times the distance d which is traveled by the geometric centroid of F.

V = a d

This looks almost the same as the formula for the surface area, above. It is so much alike you might begin to wonder why the surface areas and volumes were not always the same. The reason they both give different results is that the centroid of a plane figure is (usually) different from the centroid of its boundary curve. The area of F is also not the same as the arc length.

To go back to our example of the torus (donut) with minor radius r and major radius R, the area A will be πr2 and the distance to the centroid of F will be 2πR. So we can calculate the volume with Pappus’s second theorem.

V = (π r2) (2 π R) = 2 π2 Rr2

Stephanie Glen. "Theorem of Pappus" From Geometry for the rest of us!

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