 #### What is the surface area of a sphere?

Theorem
The surface area of a sphere, SA, is given by SA = 4πr².

Proof
Consider the following diagram. It is possible to imagine the complete sphere being made up of wedges, like in the diagram. It is also possible to approximate each wedge to a pyramid. Clearly the more wedges used, the flatter the bases become and hence the more true to pyramids they become. In addition, the smaller the bases, the closer the perpendicular height of the pyramid is to the radius of the sphere.

So the volume of each pyramid can be approximated to 1/3Akr, where Ak represents the area of the base of the k th pyramid.

Hence the volume of the sphere, V, can be approximated.

 V ≈ 1/3A1r + 1/3A2r + 1/3A3r + ... + 1/3Anr = 1/3r(A1 + A2 + A3 + ... + An)

However, as the number of wedges, n, approaches infinity, the approximation for the volume improves and the surface area of the sphere will be given by
SA = A1 + A2 + A3 + ... + An, and we already know that the volume of a sphere, V = 4πr³/3.

So we deduce that 1/3r×SA = 4π/3.

Hence SA = 4πr².