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I'm trying to calculate the height of a spherical cap on a sphere where you know the volume of the cap and the radius of the sphere.
I was able to find the equation for the volume of the cap with radius $r$ and height $h$ which is
$$V=\frac{\pi h^2}{3}(3r-h)$$
I feel really dumb but for the life of me I can't seem to rewrite the equation in such a way to find $h$ when given a known volume and radius.
Best Answer
You have to solve the cubic $$ \frac{\pi}{3}h^3-\pi r h^2+V=0 $$
that always have a real solution for $h$, and, in general, can be solved with Cardano's formula.