Euler's Formula, Proof 8: Sum of Angles

This proof uses the fact that the planar graph formed by the polyhedron can be embedded so all edges form straight line segments.

Sum up the angles in each face of a straight line drawing of the graph (including the outer face); the sum of angles in a $k$-gon is $(k-2)\pi$, and each edge contributes to two faces, so the total sum is $(2E-2F)\pi$.

Now let's count the same angles the other way. Each interior vertex is surrounded by triangles and contributes a total angle of $2\pi$ to the sum. The vertices on the outside face contribute $2(\pi -{\theta }_v)$, where ${\theta }_v$ denotes the exterior angle of the polygon at vertex $v$. The total exterior angle of any polygon is $2\pi$, so the total angle is $2\pi V-4\pi$.

Combining these two formulas and dividing through by $2\pi$, we see that $V-2=E-F$, or equivalently $V-E+F=2$.

This is the method used by Descartes in 1630. Sommerville attributes this proof to Lhuilier and Steiner. Hilton and Pederson use angles in a similar way to relate the Euler characteristic of a polyhedral surface to its total angular defect.

Proofs of Euler's Formula.
From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.