Let $t_n$ be the number of rooted 5-connected planar triangulations
with $2n$ faces. We find $t_n$ exactly for small $n$, as well as an
asymptotic formula for $n\rightarrow\infty$. Our results are found by
compositions of lower connectivity maps whose faces are triangles or
quadrangles. We also find the asymptotic number of cyclically
5-edge connected cubic planar graphs.