Outerplanar Turán number of a cycle

A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Turán number of $H$, denoted by $e{x}_{\mathcal{O}\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex outerplanar graph which does not contain $H$ as a subgraph. In 2021, L. Fang et al. determined the outerplanar Turán number of cycles and paths. In this paper, we use techniques of dual graph to give a shorter proof for the sharp upperbound of $e{x}_{\mathcal{O}\mathcal{P}}(n,C_k)\le \frac{(2k-5)(kn-k-1)}{k^2-2k-1}$.