Archimedes' Determination of the Volumes of Cones, Cylinders & Spheres

  Archimedes used the method of exhaustion to determine the volumes of a number of geometrical solids. It's considered a precursor to calculus which is used to sum infinitesimally small elements of volume, area and length. The method of exhaustion uses a series of inscribed and circumscribed geometrical solids to fill and surround a volume and focuses on the difference between the two. As the difference becomes smaller and smaller the remainder is reduced to zero. Archimedes determined that the volume of a sphere is four time that of a cone whose base is equal to the area cut by a plane through the center of the sphere and whose height is the radius of the sphere. The volume of a cone was known prior to this and can be found in Euclid's Elements to be 1/3 of the cylinder that just surrounds it. Archimedes then uses this to determine that the volume of a cylinder that just circumscribes a sphere is 3/2 that of the sphere. The ratio of the volume of a triangular prism to that of a prism that just contains it was found in Euclid to 1/3.

Some of these formulas were known even earlier as can be seen from the Moscow papyrus and the Rhind papyrus with various approximations used for the value of π.