  Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called
$\mathcal{F}$-\textit{free} if it contains no graph from $\mathcal{F}$ as a
subgraph. For a positive integer $n$, the \emph{planar Tur\'an number} of $\F$,
denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex
$\F$-free planar graph.
  Let $\Theta_k$ be the family of Theta graphs on $k\geq 4$ vertices, that is,
graphs obtained by joining a pair of non-consecutive vertices of a $k$-cycle
with an edge. Lan, Shi and Song determined an upper bound
$\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$, but for
large $n$, they did not verify that the bound is sharp. In this paper, we
improve their bound by proving $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq
\frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely
many positive integer $n$ and an $n$-vertex $\Theta_6$-free planar graph
attaining the bound.
