Sphere - Surface Area of A Sphere

Surface Area of A Sphere

The surface area of a sphere is given by the following formula:

This formula was first derived by Archimedes, based upon the fact that the projection to the lateral surface of a circumscribed cylinder (i.e. the Lambert cylindrical equal-area projection) is area-preserving. It is also the derivative of the formula for the volume with respect to r because the total volume of a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.

At any given radius r, the incremental volume (δV) is given by the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):

The total volume is the summation of all shell volumes:

In the limit as δr approaches zero this becomes:

Since we have already proved what the volume is, we can substitute V:

Differentiating both sides of this equation with respect to r yields A as a function of r:

Which is generally abbreviated as:

Alternatively, the area element on the sphere is given in spherical coordinates by . With Cartesian coordinates, the area element . More generally, see area element.

The total area can thus be obtained by integration: