Counting $k$-cycles in $5$-connected planar triangulations

We show that every $n$-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all $n\ge 20$ and this upper bound is tight. We also show that for every $k\geq 6$, there exists some constant $C(k)$ such that for sufficiently large $n$, every $n$-vertex $5$-connected planar graph has at most $C(k) \cdot n^{\lfloor{k/3}\rfloor}$ many cycles of length $k$. This upper bound is asymptotically tight for all $k\geq 6$.