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We adapt a recent theory for the random close packing of polydisperse spheresin 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 theApollonian gasket in dimensions 2 and above. Our approximate results agree withpublished values in $2$ and $3$ dimensions to within $0.05%$ and $0.6%$respectively, and we provide predictions for dimensions $4$ to $8$.

Estimate for the fractal dimension of the Apollonian gasket in d dimensions