The Surface Area Of An Ellipsoid

A. Dieckmann, Universität Bonn, July 2003

This short note shows a way to the formula for the surface area of an ellipsoid. The result is given for example in the formula gallery of the Mathematica Book (Ref. 1), but with a typo. There was no instruction accessible to me, how it is obtained, and I had to try for myself. It proved harder than I first thought, so I think it is worthwhile to put the derivation on the internet.

To make a later integration valid, we have to name the semi-axes of the ellipsoid c > a > b > 0, so that b < c and The axes may be renamed at the end.

Now calculate the area numerically – for reference:

Symmetry allows the reduction of the integration domain:

Integrate symbolically with respect to θ :

After numerical integration with respect to φ the area comes out correctly:

Express ArcSinh by ArcTanh:

And the integral can be reduced to a simpler form by substituting

The Jacobian dφ/dx is:

Check the new, 'simple' form :

Then integrate by parts with

uv(upper limit) - uv(lower limit):

There are now two integrals left to attack, but luckily, both have already been done – about 130 years ago. The first is:

If the conditions cited at the beginning

• 0 < ( t = ArcCos[b/c] ) < π/2, or equivalently b < c, and

• 0 < k < 1

are all fulfilled, then this integral has a solution, found with Gradshteyn Ryzhik (6.123) (Ref.2) to be :

The second integral is very similar to the first :

This is – with Gradshteyn Ryzhik (6.113-2) – :

Put them into the formula for the surface area above :

This is already the correct expression. Now rename the axes, that a > b > c :

The final result is ( incidentally valid for any set of semi-axes – if a ≠ b ≠ c) :

Prolate case: b→c ; surface of revolution (radius c)

Oblate case: b→a ; surface of revolution (radius a)

c→a ; surface of sphere

As a 'pocket calculator' approximation to the surface area of an ellipsoid, that is better than 1.2% (if a > b ≥ c), we may use:

This formula is shorter and has a smaller more symmetric error than the one given in a previous version of this note (2003).

Note that the first two terms correspond to the prolate surface of revolution.

References

1. Wolfram: The Mathematica Book, Wolfram Media, Inc., Fourth Edition, 1999

2. Gradshteyn/Ryzhik: Table of Integrals, Series and Products, Academic Press, Second Printing, 1981