Euler characteristic of a sphere using 'grid method'

Question

If I place the following grid (see picture) on the sphere, how can this determine its euler characteristic?

I know that the formula for the Euler characteristic for surfaces is: $E=V-E+F$ with the final 3 letters corresponding to vertices edges and faces respecitively.

I know that this is 2 for all convex polyhedron surfaces, amd thus the sphere

But how can I arrive at this value using the grid method?

Many thanks

Answer 1

I count $12$ vertices, $12$ edges, and $2$ faces (the one in the middle, as well as the "outside" — most of the sphere). $$V-E+F=12-12+2=2$$

Answer 2

In your grid, the external vertices have to be identified as one vertex to give a proper cell decomposition of the sphere. If you do this, you get $V = 5$, $E = 12$, $F = 9$ so that $V - E + F = 2$, just like Euler said.