  We adapt a recent theory for the random close packing of polydisperse spheres
in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. {\bf 131}
244104 (2009)] in order to predict the Hausdorff dimension $d_{A}$ of the
Apollonian gasket in dimensions 2 and above. Our approximate results agree with
published values in $2$ and $3$ dimensions to within $0.05%$ and $0.6%$
respectively, and we provide predictions for dimensions $4$ to $8$.
