What is the apex angle measured against? Is it a vertical line which falls from the apex perpendicular to the base? (ie: bisecting the cone in the Z axis)

The volume of a cone is which for a circular base is

We can also "draw" a triangle with the angle made with the apex being . Since this is a right-angled triangle we can use trig:

Sub this expression for into

Rearrange for h:

As ever, you should always check I've not cocked up somewhere