The Method of Mechanical Theorems - Surface Area of A Sphere

Surface Area of A Sphere

To find the surface area of the sphere, Archimedes argued that just as the area of the circle could be thought of as infinitely many infinitesimal right triangles going around the circumference (see Measurement of the Circle), the volume of the sphere could be thought of as divided into many cones with height equal to the radius and base on the surface. The cones all have the same height, so their volume is 1/3 the base area times the height.

Archimedes states that the total volume of the sphere is equal to the volume of a cone whose base has the same surface area as the sphere and whose height is the radius. There are no details given for the argument, but the obvious reason is that the cone can be divided into infinitesimal cones by splitting the base area up, and the each cone makes a contribution according to its base area, just the same as in the sphere.

Let the surface of the sphere be S. The volume of the cone with base area S and height r is, which must equal the volume of the sphere: . Therefore the surface area of the sphere must be, or "four times its largest circle". Archimedes proves this rigorously in On the Sphere and Cylinder.