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... -----