1296. Oblong and oblate

    A spheroid is an ellipsoid having two equal diameters. If we rotate an ellipse about one of its axes, we get a spheroid: an oblong spheroid if it is about the major axis and an oblate spheroid if it is about the minor axis... The volume of the oblate spheroid is greater than that of the oblong one...

    Pepe Chapuzas asked...
    
    Dear Teacher:
    If the eccentricity of an ellipse is  e , what is the ratio between the volumes of both spheroids?

    And Nina Guindilla answered...

    Dear Teacher:
    Let  a  and  b  be the major semiaxis and the minor semiaxis of the ellipse.
    Let  v  and  V  be the volumes of the oblong spheroid and that of the oblate one.

v  =  4πab²/3

V  =  4πa²b/3

v / V  =  b / a  =  √ (1−e²)