  Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not
contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of
"extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016)
213--230], that is, how many edges can an $\mathcal{F}$-free planar graph on
$n$ vertices have? We define $ex_{_\mathcal{P}}(n,\mathcal{F})$ to be the
maximum number of edges in an $\mathcal{F}$-free planar graph on $n $ vertices.
Dowden obtained the tight bounds $ex_{_\mathcal{P}}(n,C_4)\leq15(n-2)/7$ for
all $n\geq4$ and $ex_{_\mathcal{P}}(n,C_5)\leq(12n-33)/5$ for all $n\geq11$. In
this paper, we continue to promote the idea of determining
$ex_{_\mathcal{P}}(n,\mathcal{F})$ for certain classes $\mathcal{F}$. Let
$\Theta_k$ denote the family of Theta graphs on $k\ge4$ vertices, that is,
graphs obtained from a cycle $C_k$ by adding an additional edge joining two
non-consecutive vertices. The study of $ex_{_\mathcal{P}}(n,\Theta_4)$ was
suggested by Dowden. We show that $ex_{_\mathcal{P}}(n,\Theta_4)\leq12(n-2)/5$
for all $n\geq 4$, $ex_{_\mathcal{P}}(n,\Theta_5)\leq5(n-2)/2$ for all $n\ge5$,
and then demonstrate that these bounds are tight, in the sense that there are
infinitely many values of $n$ for which they are attained exactly. We also
prove that $ex_{_\mathcal{P}}(n,C_6)\le ex_{_\mathcal{P}}(n,\Theta_6)\le
18(n-2)/7$ for all $n\ge6$.
