This is a follow-up to a previously posted message by Dr. Sandy
Stewart on the surface area of an ellipsoid. This was a very
interesting posting since what would intutively seem to be a
rather simple problem (since an ellipsoid is such a simple
object) was not simple at all.

The attached file (a 2-page PDF document) gives a short derivation
of the definite integral for the surface area of an ellipsoid.
You will need Reader 3.0 to display/print it which is free and can be
downloaded from This definite integral can
be used to obtain the surface area of a general (triaxial) ellipsoid:

2 2 2
x + y + z
--- --- --- = 1
2 2 2
a b c

for any a,b,c > 0. The actual numerical integration is easily
accomplished in Mathematica with the precision of the
estimated area chosen by the user.

Also, it may be reassuring to know that the last symbolic
formula given (surface area of an asymmetric ellipsoid) in
Dr. Stewart's summary is quite likely correct -- at least I
was able to find the same equation. However, two warnings in
using this equation: 1) it can be numerically unstable when
the semi-axes (a,b,c) or close (in value) even if a > b > c,
2) the elliptic integrals given in some references can have
opposite ordering to those shown in the symbolic equation. Also,
it is interesting that after extensive testing of the numercial
evaluation of the definite integral and direct evaluation of
the symbolic equation it seems that the numerical evaluation
is more accurate!


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V. P. Stokes *
BMC, Neuroscience *
Karolinska Institute Quality Has No Fear *
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