Pappus's centroid theorems are results from geometry about the surface area and volume of solids of revolution. These quantities can be computed using the distance traveled by the centroids of the curve and region being revolved.
Let be a curve in the plane. The area of the surface obtained when is revolved around an external axis is equal to the product of the arc length of and the distance traveled by the centroid of
Let be a region in the plane. The volume of the solid obtained when is revolved around an external axis is equal to the product of the area of and the distance traveled by the centroid of
Consider the cylinder obtained by revolving a rectangle with horizontal side and vertical side around one of its vertical sides (say its left side). The surface area of the cylinder, not including the top and bottom, can be computed from Pappus's theorem since the surface is obtained by revolving its right side around its left side. The arc length of its right side is and the distance traveled by its centroid is simply so its area is
The volume of the cylinder is the area of the rectangle multiplied by the distance traveled by its centroid. The centroid of the rectangle is its center, which is a distance of from the axis of revolution. So it travels a distance of as it revolves. The volume of the cylinder is
To compute the volume of a solid formed by rotating a region around an external axis (a similar argument applies for surface area), one can break the region up into small regions of area that are located a distance from the axis. Since these regions travel a distance when revolved around the axis, their contribution to the volume of the solid is roughly Adding up all the contributions gives the volume (as usual, passing to the limit where becomes and the sum becomes an integral, makes this an exact computation).
The difficulty is that all the different small regions are different distances away from the axis. To make this method computationally feasible, one would need to know the average distance of all the small pieces, so that the volume of the region is times the area This is precisely what Pappus' centroid theorem gives: it identifies as the distance from the centroid of the region to the axis of revolution.
Of course, this does not make the computation trivial in general, since computing the centroid of a region (or curve) is not easy, even for relatively simple shapes. However, there are sometimes symmetry considerations or other computational aids that make Pappus' theorem an effective shortcut for solving problems involving volumes and areas of revolution.
The surface area and volume of a torus are quite easy to compute using Pappus' theorem. A torus is a circle of radius centered at and rotated around the -axis. The centroid of both the surface of the circle and the region enclosed by the circle is just the center of the circle. This travels a distance of when it revolves, so the surface area is times the arc length of the circle, and the volume is times the area of the circle. That is,
Revolving a right triangle with legs of length and around the leg of length produces a cone. The surface of the cone (not including the circular base) is obtained by revolving the hypotenuse around that leg. The centroid of the hypotenuse is just the midpoint, located halfway up the side of the cone, which travels a distance as it rotates. So the surface area is by Pappus' theorem.
The centroid of the triangle is the center of mass of the three vertices (see the Triangle Centroid wiki), which is located at a distance of from the axis of revolution and above the base So the volume is times the area of the triangle, which is The product is the familiar formula for the volume of a cone.
The volume and surface area of a sphere are computable using Pappus's theorems, but the computations involve nontrivial integrals; Pappus's theorems do not provide a "shortcut" in this case.
Let be the curve The sphere of radius is obtained by revolving around the -axis. The arc length of is just since it is half a circle.
The centroid of is on the -axis by symmetry. Its -coordinate is given by the formula
The reasoning here is that the centroid is a center of mass and the mass of a small piece of the arc over a small length on the -axis is proportional to the arc length, which is (See the Arc Length wiki for a thorough explanation.) So the location of the centroid is
So the surface area is times the -coordinate of the centroid, times the arc length, which is
The -coordinate of the centroid of the disk is the (double) integral of over the region, divided by the area of the region:
So the volume is times the -coordinate of the centroid, times the area, which is