  We study the topic of "extremal" planar graphs, defining
$\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible
in a planar graph on $n$ vertices that does not contain a given graph $H$ as a
subgraph. In particular,we examine the case when $H$ is a small cycle,obtaining
$\mathrm{ex_{_{\mathcal{P}}}}(n,C_{4}) \leq \frac{15}{7}(n-2)$ for all $n \geq
4$ and $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{5}) \leq \frac{12n-33}{5}$ for all $n
\geq 11$, and showing that both of these bounds are tight.
