  The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of
edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph.
The topic of extremal planar graphs was initiated by Dowden (2016). He obtained
sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$.
Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$
and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this
paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over
7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns
out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$,
the maximum number of edges in an $n$-vertex planar graph which does not
contain $K_4$ or $C_7$ as a subgraph.
