In this blog, I used polar coordinates to derive the well-known expression for the area of a circle, . In today’s blog, I will go from 2 to 3-dimensions to derive the expression for the surface area of a sphere, which is . To do this, we need to use the 3-dimensional equivalent of polar coordinates, which are called spherical polar coordinates.

Let us imagine drawing a line in 3-D space of length into the positive part of the coordinate system. We will draw this line at an angle above the x-y (horizontal) plane, and at an angle to the y-z (vertical) plane (see the figure below). When we drop a vertical line from our point onto the x-y plane it has a length , as shown in the figure below.

We then increase the angle by a small amount , and increase the angle by a small amount . As the figure shows, the small surface element which is thus created is just multiplied by , so .

To find the surface area of the sphere, we need to integrate this area element over the entire surface of the sphere. Therefore, we keep and we vary . We can go from a (the negative to the positive ), and from a (one complete rotation about the z-axis on the x-y plane), so we have

so, the total surface area of a sphere is

as required. In a future blog I will use spherical polar coordinates to derive the volume of a sphere, where will no longer be constant as it is here.