We all live on an oblate spheroid , so it could be handy to know a little about oblate spheroids.
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.
The volume of an oblate spheroid is simple:
Clearly we recover the volume of a sphere when a = c.
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 √((a² + c²)/2), with error on the order of e².
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
 Not in a yellow submarine as has been suggested.
2 thoughts on “Geometry of an oblate spheroid”
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?
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.