proper shere cap angle

Question

if I have a sphere and I want to cut a chunk off the top making a sphere cap, at what angle from the top of the sphere cap down, would create a cap giving x% of the total volume of the sphere, (see wikipedia Sphereical cap)

the volume of a sphere is V=4*pi*R^2/3

the volume of a sphere cap is v=Pi*h^2*(3*r-h)/3

with h=sin(theta);

therefore v = 1/3*Pi*sin(theta)^2*(3*r-sin(theta))

and x=v/V

therefore x=(1/3*Pi*sin(theta)^2*(3*r-sin(theta)))/(4*pi*R^2/3)

is there a good way to solve for theta in terms of x?

Answer 1

Well, the volume of a sphere is (4/3)(pi)r^3 not 4*pi*R^2/3. What else might be wrong?