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.