Euler's formula for planar graphs

Statement

Let $G$ be a finite connected planar graph, drawn in the plane with no crossing edges. Let $v$, $e$ and $f$ be the numbers of vertices, edges and faces of the drawing, where the unbounded face counts. Then $v - e + f = 2$.

Remarks

Proof idea: if the graph has a cycle, delete an edge of the cycle. This removes one edge and one face. At the end you have a spanning tree, with $e = v - 1$ and $f = 1$. The same formula holds for the vertices, edges and faces of a convex polyhedron.

Explanations

L1 human

Draw a connected graph without crossings and count its vertices, edges and regions. Vertices minus edges plus regions is always $2$. For a square, this is $4 - 4 + 2$.