On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC.^{[1]} It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.^{[2]}
The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. In his work, Archimedes showed that the surface area of a cylinder is equal to:
and that the volume of the same is:
On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to:
The result for the volume of the contained balled stated that it is twothirds the volume of a circumscribed cylinder, meaning that the volume is
Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation.^{[4]}
The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which, of course, did not exist in Archimedes' time. Archimedes used an inscribed halfpolygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, which he then determined the volume of.^{[5]}
