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

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 crosssections are all circles with radii SR, SP, and SN, respectively. What Archimedes discovered was that if the crosssections 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.