You are here

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

Gabriela R. Sanchis

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

Volume Center of Gravity
Cylinder \(\pi\)r2h
On the cylinder's axis, half-way between top and bottom
Cone (\(\pi\)r2h)/3
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.

Cylinder, sphere, and cone

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.

Gabriela R. Sanchis, "Archimedes' Method for Computing Areas and Volumes - Cylinders, Cones, and Spheres," Convergence (June 2010)