# How to calculate excluded volume in Onsager's hard-rod model?

Can somebody please provide a derivation of how to calculate excluded volume of two rods with angle of intersection being $\gamma$. rods are cylinders, capped with semi-spheres. Onsager theory of hard rods is based on this and I cant seem to find a derivation for excluded volume.

In order to calculate the excluded volume between two spherocylindric rods, the relative angle between them changes the base area of the excluded volume, usually taken as a parallelogram, and the thickness of rods, i.e. $D$ the height of the excluded volume. Consider the picture below, taken from (Basic Concepts for Simple and Complex Liquids, from Jean-Louis Barrat, Jean-Pierre Hansen)
As you can see, we have a base in the form of a parallelogram with area $L^2 \sin\gamma$ and with thickness $2D$, excluded volume: $$V_{ex}=2L^2D|\sin \gamma|.$$
The $2$ comes from the fact that the volume $L^2D|\sin \gamma|$ is excluded for the other rod on both sides.