By: Tao Steven Zheng（郑涛）

【Problem】

In Volume I of On the ''Sphere and the Cylinder'', Archimedes (c. 287 - 212 BC) determined the volumetric ratio of a sphere to a circumscribed cylinder. The height and width of the cylinder is equal to the diameter of the sphere. What is this ratio?

【Solution】

Let the radius of the sphere be . The circumscribed cylinder shares the same height and width as the sphere, so the height of the cylinder is .

The volume of a sphere is , and the volume of a cylinder is . Thus, the volume circumscribed cylinder is

Therefore, the volumetric ratio of a sphere to its circumscribed cylinder is

which simplifies to