Wikipedia has the following formula:
V = πh(3a2+ h2) / 6
... for the volume of the cap given the base radius a and cap height h. Solving this for h requires solving a cubic equation:
h^3 + 3ah - 6V/π = 0
I don't see an easy way to factor this.
WolframAlpha gives this gory solution for the only real root:
x = (sqrt(pi^2 a^3+9 V^2)+3 V)^(1/3)/pi^(1/3)-(pi^(1/3) a)/(sqrt(pi^2 a^3+9 V^2)+3 V)^(1/3)
I had to rename the unknown from h to x to get Alpha to recognize it as a cubic to solve. Anyway, if Wikipedia is right, and WolframAlpha is accurate, any other approach will give an answer that is equivalent to this. So, either there's no easy answer, or this is half of a truly impressive identity.
jesus..though question dudee