The surface area of a prolate ellipsoid can be approximated as 4πAB under certain conditions, particularly when comparing it to the spherical surface area formula 4πR². This approximation holds true when substituting for the semi-major axis 'a' in terms of the semi-minor axis 'c' and the eccentricity 'e', as detailed in equations 8 and 9 from the Wolfram MathWorld resource. The exact formula for the surface area is given by 2πa² + (2πac sin⁻¹(e))/e, highlighting the dependency on the eccentricity for accuracy.
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Look at eqn 9 at http://mathworld.wolfram.com/ProlateSpheroid.html.Tahmeed said:Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??
I don't know why but the link isn't working for me. However does it bring 4piAC as a result?haruspex said:Look at eqn 9 at http://mathworld.wolfram.com/ProlateSpheroid.html.
You could substitute for a in terms of c and e using eqn 8, then make an approximation for small e.
It can't be exactly that, of course, so it depends how good an approximation you want.Tahmeed said:I don't know why but the link isn't working for me. However does it bring 4piAC as a result?