  The Apollonian packings (APs) are fractals that result from a space-filling
procedure with spheres. We discuss the finite size effects for finite intervals
$s\in[s_\mathrm{min},s_\mathrm{max}]$ between the largest and the smallest
sizes of the filling spheres. We derive a simple analytical generalization of
the scale-free laws, which allows a quantitative study of such \textit{physical
fractals}. To test our result, a new efficient space-filling algorithm has been
developed which generates random APs of spheres with a finite range of
diameters: the correct asymptotic limit
$s_\mathrm{min}/s_\mathrm{max}\rightarrow 0$ and the known APs' fractal
dimensions are recovered and an excellent agreement with the generalized
analytic laws is proved within the overall ranges of sizes.
