where x is the distance along the axis from the bottom ($x=0$) to the top ($x=1$) and y is the distance from the axis. The envelope of this family of lines is the hyperbola $$ y^2=\sin^2(\alpha/2)(2x-1)^2+\cos^2(\alpha/2)\tag{3} $$ The volume of the hyperboloid of revolution is pretty simple to compute $$ \begin{align}