how do I calculate the Euler characteristic of a Klein bottle?

Question

I guess I have to use that it is a closed surface (manifold) and therefore its boundary is empty, so then I gotta use Gauss-Bonnet easily (but I just cannot think of how to use it.)

i.e. $$\chi(\textit{Klein Bottle})=\frac{1}{2\pi}\int\int_{\textit{Klein Bottle}}K\,d\sigma$$

I was thinking on using that a Klein bottle is homeomorphic to 2 "glued" Möbius strips, but how from then...?

P.S. this is from my first course on Differential Geometry, so talkings on higher-level courses wouldn't help me at all

Answer 1

This is one triangulation of the Klein bottle, with vertex $V$ and edges $a, b, c$:

Can you calculate the Euler characteristic from this?