We all live on an oblate spheroid [1], so it could be handy to know a little about oblate spheroids.

## Eccentricity

Conventional notation uses *a* for the equatorial radius and *c* for the polar radius. Oblate means *a* > *c*. The eccentricity *e* is defined by

For a perfect sphere, *a* = *c* and so *e* = 0. The eccentricity for earth is small, *e* = 0.08. The eccentricity increases as the polar radius becomes smaller relative to the equatorial radius. For Saturn, the polar radius is 10% less than the equatorial radius, and *e* = 0.43.

## Volume

The volume of an oblate spheroid is simple:

Clearly we recover the volume of a sphere when *a* = *c*.

## Surface area

The surface area is more interesting. The surface of a spheroid is a surface of rotation, and so can easily be derived. It works out to be

It’s not immediately clear that we get the area of a sphere as *c* approaches *a*, but it becomes clear when we expand the log term in a Taylor series.

This suggests that an oblate spheroid has approximately the same area as a sphere with radius √((*a*² + *c*²)/2), with error on the order of *e*².

If we set *a* = 1 and let *c* vary from 0 to 1, we can plot how the surface area varies.

Here’s the corresponding plot where we use the eccentricity *e* as our independent variable rather than the polar radius *c*.

## More differential geometry posts

[1] Not in a yellow submarine as has been suggested.

Pardon me if I’m wrong. I’m here to be intellectually stretched.

For a sphere then e is zero and the second term in the first area equation would get log(1). That’s also zero and cancels the second term, leaving the usual results for spheres because a is r. True?

Phil,

In the first equation for area, you can’t let e -> 0 directly because the log term goes to zero but the 1/e term goes to infinity.

In the second equation for area, you can let e-> 0 and you get 4 pi r^2, since e -> 0 implies a and c converge to a common r.

You could use L’Hopital’s rule to take the limit in the first equation and you’d end up at the same place.