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