How do you find the area of an oblate spheroid?

Answer 1

An oblate spheroid has volume since it is a three dimensional object. I assume that is what you meant.

The volume of an ellipsoid with radii along the three axes of a, b, and c is:

(4/3)πabc

Now an oblate spheroid is a special case of an ellipsoid. It can be generated by rotating an ellipse about one of its axes. So two of the radii will be the same. The volume therefore is:

(4/3)πa²b

If you were looking for surface area, that is another more difficult question.

Answer 2

A reasonable approximation would be A = 4 pi (Re^2 x Rp)^(2/3), where Re is the equatorial radius and Rp is the polar radius. I'm pretty sure that the exact formula involves elliptic integrals so cannot be computed exactly in closed form.

Answer 3

Elementary area ds = 2pix * sqrt((dx)^2 +(dy)^2) = 2pix * sqrt(1 +(y’)^2) *dx, where (x/a)^2 +(y/b)^2 =1 is an ellipse rotated around y-axis;
Or x=a*cos(t), y=b*sin(t) is ellipse in parametric form; thus ds = 2pi*a*cost * sqrt((a*sint)^2 +(b*cost)^2)*dt;
Now S= 4pi*a*∫cost*sqrt(((a*sint)^2 +(b*cost)^2)*dt {for t=0 until pi/2};
Use numeric method of integration.
If a=b=r, then S=4pi*r^2 for correct sphere.