  We investigate the minimum number of cycles of specified lengths in planar
$n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for
any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil
(\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius
of the triangulation's dual, which is at least logarithmic but can be linear in
the order of the triangulation. We also show that there exist planar
hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any
$k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove
that planar 4-connected $n$-vertex triangulations contain $\Omega(n)$ many
$k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain
additional conditions, they contain $\Omega(n^2)$ $k$-cycles for many values of
$k$, including $n$.
