Height of a spherical cap from volume and radius $\hskip 2.2 in$

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.
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.
Given $V$ in order to find $h$ we have to find roots of the cubic equation.
EDIT1:
It is more convenient to express given volume as a fraction$f$ of the full sphere$ \;4\pi r^3/3$
$$\dfrac{\pi h^2(3r-h)}{3}=f \dfrac43 \pi r^3 \;\;\rightarrow 4fr^3+h^3-3r h^2=0$$
The following is given by Mathematica for $h$ value.
  Solve[-3 r h^2 + h^3 + 4 f r^3 == 0, h]

$$ \left(r+\frac{1}{2} \left(1+i \sqrt{3}\right) \sqrt[3]{2 \sqrt{f^2 r^6-f r^6}-2 f r^3+r^3}+\frac{\left(1-i \sqrt{3}\right) r^2}{2 \sqrt[3]{2 \sqrt{f^2 r^6-f r^6}-2 f r^3+r^3}}\right)
$$
in which the imaginary part cancels on evaluation.
Else if you want a graphic solution read off $h$ for given $ (V,r)$  or vice-versa as shown: