A spheroid is bisected into two spheroidal caps by a plane, such that the shape of the area of the plane inside the spheroid is elliptical. The alignment of the plane is defined by two angles theta1 and theta2, which are the angles made between the plane and the normal planes to the spheroid at the two points at either end of the major axis of the ellipse. I know the volume V of the spheroidal cap. However I want to find the length of the major axis of the ellipse. So I need an equation for the length of this major axis in terms of V, theta1 and theta2.
How can I derive such an equation please? I can't embed an image as I'm a newbie but this might hopefully illustrate the scenario a little more clearly, it shows a cross-section through the middle of the spheroid, with the length a being the major axis of the ellipse.
Grateful for any help.