Planar graphs with the maximum number of induced 4-cycles or 5-cycles

For large $n$ we determine exactly the maximum numbers of induced $C_4$ and $C_5$ subgraphs that a planar graph on $n$ vertices can contain. We show that $K_{2,n-2}$ uniquely achieves this maximum in the $C_4$ case, and we identify the graphs which achieve the maximum in the $C_5$ case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Gy\H{o}ri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.