  Finding the maximum number of induced cycles of length $k$ in a graph on $n$
vertices has been one of the most intriguing open problems of Extremal Graph
Theory. Recently Balogh, Hu, Lidick\'{y} and Pfender answered the question in
the case $k=5$. In this paper we determine precisely, for all sufficiently
large $n$, the maximum number of induced $5$-cycles that an $n$-vertex planar
graph can contain.
