The packing density of an arrangment of circles on a flat surface, denoted by η, is the proportion of the surface area covered by the arranged circles.
In 1773, Joseph Louis Lagrange proved that the lattice arrangement of circles with the highest possible packing density is the hexagonal packing arrangement, in which the centers of the circles are arranged in a hexagonal lattice such that each circle is surrounded by 6 other circles:
The packing density of this arrangement is η = π / √12 ≈ 0.9069.
Given that the most efficient way to pack differently sized circles together is not obvious, the table's η value is close to but less than 0.9069:
