That formula cannot be true.  Consider an ellipsoid with a = b.  Then L_ac = L_bc are given by an EllipticE, while L_ab = 2πa.  If the conjectured formula were true, we would be able express EllipticE in terms of elementary functions.   --  Gene On Friday, June 21, 2019, 3:31:50 PM PDT, Dan Asimov <dasimov@earthlink.net> wrote: The actual question Bill asked there was how to figure the volume of an ellipsoid from its 3 principal circumferences (call them L_ab, L_bc, L_ca). That's a quite interesting question, because arclength of an ellipse is notoriously not in some nice closed form with elementary functions, yet there still might be some simple formula of form   F(L_ab, L_bc, L_ca)  =  (4π/3)abc. where F should be symmetric in its 3 variables. Hmm, since the L's are 1-dimensional and volume is 3 dimensional, maybe just (*)  K * L_ab * L_bc * L_ca  =  (4π/3)abc. Wouldn't that be nice! If so, then for the unit sphere we get   K * (2π)^3  =  4π/3, or   K  =  1/(6 π^2) — IF (*) is correct in general. Now I'd like to test (*) on a noncircular ellipse whose exact semi-axes and circumference are known in simple terms. But maybe other people know better how to find such examples. (Needless to say, refuting (*) would require only a good numerical value.) —Dan Bill Gosper wrote: ----- https://math.stackexchange.com/questions/3268714/what-are-the-semiaxes-a-b-c... -----