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On the Sphere and Cylinder

In the treatise by this name, written c. 225 BCE, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is 4 / 3 π r = 2 / 3 (2 π r). The surface area of this sphere is 4 π r = 2 / 3 (6 π r). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.