I have been reading an old draft that I made a few years ago where I used the following expression for the area of an ellipsoid:
$$A=2 \pi r^2 + \frac{2 \pi \tanh^{-1}(\sqrt{1-r^{-6}})}{r^4 \sqrt{1- r^{-6}}}$$
I considered an ellipsoid defined by two parameters, the height and the radius ($r$). I assume that the volume of the ellipsoid is equal to $4\pi/3$ obtaining $h=1/r^2$. I haven't been able to find this equation anywhere and I forgot to write down the reference where I found it.
Where does this equation come from? How does it relate to the usual formula for the area of an ellipsoid?
This comes from the formula for the surface area of an oblate spheroid (ie, an ellipsoid formed by spinning an ellipse about its minor axis). $$S = 2\pi a^2 \left(\;1 + \frac{1-e^2}{e}\;\operatorname{atanh} e \;\right) \qquad\text{where}\qquad e^2 = 1 - \frac{c^2}{a^2}$$ where $a$ is the semi-major axis and $c$ the semi-minor axis.
Specifically, taking $a = r$ and $c = r^{-2}$ (your $h$), we get $e^2 = 1 - r^{-6}$, so that $$S = 2\pi r^2\left(\; 1 + \frac{\operatorname{atanh} \sqrt{1-r^{-6}}}{r^6\;\sqrt{1 - r^{-6}}}\;\right)$$