Triangular lattice packing of discs (${\varphi }_{\mathrm{cp}}=\pi /(2\sqrt{3})$)

A triangular array with nearest-centre spacing $2r$ has primitive cell area $2\sqrt{3}{r}^2$. Dividing the disc area $\pi r^2$ by that area gives packing fraction $\pi /(2\sqrt{3})\simeq 0.9069$.