Theorem of Pappus

Pappus’s Theorem for volume connects the geometry of a plane region with the volume of its solid of revolution, without using integration techniques like the disk method.

The theorem states that when a plane region with a known centroid rotates around an axis, the resulting volume equals the region's area times the distance traveled by its centroid along the circular path.

For example, consider a three-dimensional shape that looks like a doughnut, called a torus. It is formed by rotating a circle of radius r around a line in its plane that does not intersect the circle.

This circular cross-section generates the solid during rotation.

The circle’s centroid lies at a distance R from the rotation axis.

As the circle rotates, its centroid follows a circular path with a length equal to its circumference.

Using the theorem, the volume of the torus is found by multiplying the cross-sectional area by the distance traveled by the centroid.

A practical example is the use of toroidal tanks to store fuel in tight spaces. Pappus’s Theorem uses the area and the centroid’s path length to find the volume, which helps show the available fuel capacity.