Given a family , a graph is -free if it does not contain any graph in as a subgraph. We continue to study the topic of “extremal” planar graphs initiated by Dowden (2016), that is, how many edges can an -free planar graph on vertices have? We define to be the maximum number of edges in an -free planar graph on vertices. Dowden obtained the tight bounds for all and for all . In this paper, we continue to promote the idea of determining for certain classes . Let denote the family of Theta graphs on vertices, that is, graphs obtained from a cycle by adding an additional edge joining two non-consecutive vertices. The study of was suggested by Dowden. We show that for all , for all , and then demonstrate that these bounds are tight, in the sense that there are infinitely many values of for which they are attained exactly. We also prove that for all .