A while ago there was a question about how the surface area of an ellipse
changes with the perimeter... I've been talking to a friend and here's
his response: (Hope this helps!)


Sorry to take so long to respond. I asked a few people about the
surface area of the ellipsoid problem, and apparently you need elliptic integral
s to solve it. I think there's books which have tables of them, or I'm
sure mathematica could solve them explicitly. As far as a derivation, I think
it's straightforward but complicated. It sounds like your friend knows more
about it than me, though.