Volume fraction of Hexagonally-Closed-Packed (HCP) Circles

First, we know that the tightest way to stack circles in 2D is like so:

This is called Hexagonally Close Packed because we can imagine space-filling hexagons overlapping these circles with the vertices of the hexagons at the centers of each circle.

Zooming in on a portion of this hexagon, we can see that here are triangles that are also space filling and can represent a unit-cell from which we can calculate the volume fraction: