Let the radius of the top of the glass be $R$, and let the bottom radius be $r$, where $0. Let the height of the glass be $h$. The glass can be extended to a cone. We will be working with several cones, all similar to each other. Since I do not remember the formula for the volume of a cone, we use scaling arguments.
Imagine a cone similar to our cones, but with top radius $1$. Choose units of volume so that this top radius $1$ cone has volume $1$. (That's OK, the unit of volume need not be simply related to the unit of length. There exists a country where distances are measured in feet but volumes in gallons.)
If we extend our glass to a complete cone, the volume of that complete cone is, by scaling, $R^3$. Similarly, the part of that cone which is beyond the glass has volume $r^3$. So the glass itself has volume $R^3-r^3$. Half of this is $(R^3-r^3)/2$. Add back the missing $r^3$ which lies beyond the glass. We conclude that the cone made up of the bottom half of the orange juice plus the stuff beyond the glass has volume $\frac{R^3-r^3}{2}+r^3\quad\text{or more simply}\quad \frac{R^3+r^3}{2}.$
It follows that when exactly half of the orange juice is removed from the glass, the top of the orange juice is a circle of radius $\sqrt[3]{\frac{R^3+r^3}{2}}.$ Pouring out juice until radius is just right sounds awkward. So let's work with heights. A similar triangle argument shows that the height of the "missing" cone is $\frac{hr}{R-r}.$ By scaling it follows that the height of the half-full of juice cone is $\frac{h}{R-r}\sqrt[3]{\frac{R^3+r^3}{2}}.$ To get the actual orange juice height when the glass is half-full, subtract the height of the missing cone. We get $\frac{h}{R-r}\left(\sqrt[3]{\frac{R^3+r^3}{2}}-r\right).$
Remark: Scaling arguments are somewhat underused in elementary mathematics. Physicists use them much more routinely.