Density is measured by the fraction of area covered by the discs.
For example, the density of the layout on the left below is the ratio of the area of a circle to the square which just encloses it, and the density of the layout on the right is the ratio of the area of a circle to that of its circumscribing regular hexagon. It is visibly apparent, and easy to calculate explicitly, that the density on the right is greater. The density of a lattice packing is the ratio of the area of a disc to that of a fundamental parallelogram, and among all lattice packings with a given size disc the hexagonal lattice clearly minimizes the area of a suitable fundamental parallelogram. This was observed first by Gauss. The fundamental parallelogram of this lattice is a square, and the density of the distribution of discs is the ratio of the area of a circle to its circumscribing square.