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.