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   Surface Ellipsoid_1.gif The axes may be renamed at the end.

Surface Ellipsoid_2.gif

Surface Ellipsoid_3.gif

Now calculate the area numerically – for reference:

Surface Ellipsoid_4.gif

Surface Ellipsoid_5.gif

Symmetry allows the reduction of the integration domain:

Surface Ellipsoid_6.gif

Surface Ellipsoid_7.gif

Integrate symbolically with respect to θ :

Surface Ellipsoid_8.gif

Surface Ellipsoid_9.gif

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

Surface Ellipsoid_10.gif

Surface Ellipsoid_11.gif

Express ArcSinh by ArcTanh:

Surface Ellipsoid_12.gif

Surface Ellipsoid_13.gif

Surface Ellipsoid_14.gif

And the integral can be reduced to a simpler form by substituting  Surface Ellipsoid_15.gif

Surface Ellipsoid_16.gif

Surface Ellipsoid_17.gif

Surface Ellipsoid_18.gif

The Jacobian dφ/dx is:

Surface Ellipsoid_19.gif

Surface Ellipsoid_20.gif

Check the new, 'simple' form :

Surface Ellipsoid_21.gif

Surface Ellipsoid_22.gif

Then integrate by parts with  Surface Ellipsoid_23.gif

Surface Ellipsoid_24.gif

Surface Ellipsoid_25.gif

Surface Ellipsoid_26.gif

Surface Ellipsoid_27.gif

Surface Ellipsoid_28.gif

Surface Ellipsoid_29.gif

Surface Ellipsoid_30.gif

Surface Ellipsoid_31.gif

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

Surface Ellipsoid_32.gif

Surface Ellipsoid_33.gif

Surface Ellipsoid_34.gif

Surface Ellipsoid_35.gif

Surface Ellipsoid_36.gif

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

Surface Ellipsoid_37.gif

Surface Ellipsoid_38.gif

Surface Ellipsoid_39.gif

Surface Ellipsoid_40.gif

Surface Ellipsoid_41.gif

Surface Ellipsoid_42.gif

Surface Ellipsoid_43.gif

Surface Ellipsoid_44.gif

Surface Ellipsoid_45.gif

Surface Ellipsoid_46.gif

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 :

Surface Ellipsoid_47.gif

Surface Ellipsoid_48.gif

The second integral is very similar to the first :

Surface Ellipsoid_49.gif

Surface Ellipsoid_50.gif

Surface Ellipsoid_51.gif

Surface Ellipsoid_52.gif

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

Surface Ellipsoid_53.gif

Surface Ellipsoid_54.gif

Put them into the formula for the surface area above :

Surface Ellipsoid_55.gif

Surface Ellipsoid_56.gif

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

Surface Ellipsoid_57.gif

Surface Ellipsoid_58.gif

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

Surface Ellipsoid_59.gif

Surface Ellipsoid_60.gif

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

Surface Ellipsoid_61.gif

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

Surface Ellipsoid_62.gif

c→a ; surface of sphere

Surface Ellipsoid_63.gif

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:

Surface Ellipsoid_64.gif

Surface Ellipsoid_65.gif

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.


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

Spikey Created with Wolfram Mathematica 8.0       Go to Download Page