  Given a finite simple triangulation, we estimate the sizes of circles in its
circle packing in terms of Cannon's vertex extremal length. Our estimates
provide control over the size of the largest circle in the packing. We use
them, combined with results from [12], to prove that in a proper circle packing
of the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller
and Sheffield, the size of the largest circle goes to zero with high
probability.
