Archimedes' Method for Computing Areas and Volumes - Cylinders, Cones, and Spheres

Author(s):
Gabriela R. Sanchis

Recall the following information about cylinders and cones with radius r and height h:

 Solid Volume Center of Gravity Cylinder $\pi r^2 h$ On the cylinder's axis, half-way between top and bottom Cone $\frac13\pi r^2 h$ On the cone's axis, three times as far from the vertex as from the base

Suppose a sphere with radius r is placed inside a cylinder whose height and radius both equal the diameter of the sphere. Also suppose that a cone with the same radius and height also fits inside the cylinder, as shown below.

We place the solids on an axis as follows:

For any point S on the diameter AC of the sphere, suppose we look at a cross section of the three solids obtained by slicing the three solids with a plane containing point S and parallel to the base of the cylinder. The cross-sections are all circles with radii SR, SP, and SN, respectively. What Archimedes discovered was that if the cross-sections of the cone and sphere are moved to H (where |HA| = |AC|), then they will exactly balance the cross section of the cylinder, where HC is the line of balance and the fulcrum is placed at A.

This is not hard to show. If the radius of the sphere is $r$, the origin is at $A$, and the $x$ coordinate of $S$ is $x$, then the cross-section of the sphere has area $\pi(r^2-(x-r)^2)=\pi(2r x-x^2)$, the cross-section of the cone has area $\pi x^2$, and the cross-section of the cylinder has area $4\pi r^2$. So according to the law of the lever, in order for the above balancing relationship to hold we need to following equation to be true: $2r\left[\pi x^2+\pi(2r x-x^2)\right]=4\pi r^2 x$ which can easily be verified.

Gabriela R. Sanchis, "Archimedes' Method for Computing Areas and Volumes - Cylinders, Cones, and Spheres," Convergence (June 2016)