  The one-body free volume, which determines the entropy of a hard disk system,
has extensive (cavity) and intensive (cell) contributions. So far these
contributions have not been unified and considered separately. The presented
theory incorporates both contributions, and their sum is shown to determine the
free volume and partition function. The approach is based on multiple
intersections of the circles concentric with the disks but of twice larger
radius. The result is exact formulae for the extensive and intensive entropy
contributions in terms of the intersections of just two, three, four, and five
circles. The method has an important advantage for applications in numerical
simulations: the formulae enable one to convert the disk coordinates into the
entropy contribution directly without any additional geometric construction.
The theory can be straightforwardly applied to a system of hard spheres.
