Edge bound for planar graphs

Statement

Let $G$ be a finite simple connected planar graph with $v \ge 3$ vertices and $e$ edges. Then $e \le 3v - 6$.

Remarks

Proof idea: each face is bounded by at least $3$ edges, and each edge touches at most $2$ faces. Thus $3f \le 2e$, and Euler's formula gives the bound. If $G$ also has no cycle of length $3$, then $e \le 2v - 4$. A consequence: each finite simple planar graph has a vertex of degree at most $5$.

Explanations

L1 human

A planar graph cannot have too many edges: at most about three times its number of vertices. This simple count already shows that some graphs, such as $K_5$, are not planar.