Spherical-triangle proof of the polyhedron Euler formula ($V-E+F=2$)

Radially project a convex polyhedron from an interior point onto the unit sphere and triangulate each face without adding vertices. There are $T=2E-2F$ spherical triangles. The Gauss-Bonnet theorem gives each area as angle sum minus $\pi$. Summing angles around the projected vertices gives $2\pi V$, while the total area is $4\pi$. Thus $4\pi =2\pi V-\pi T$, proving the formula. Convexity guarantees that the projected triangles cover the sphere without overlaps of interiors.