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History

The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus. The volume and area formulas were first determined in Archimedes 's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.

Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus. A similar problem – to construct a segment equal in volume to a given segment, and in surface to another segment – was solved later by al-Quhi.