the Euler characteristic of $K\#K$

Question

I'm trying to find the Euler characteristic of $K\#K$, here $K$ is the Klein bottle. I tried to use the fact that $\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$ but it seems not working hence I'm thinking about using other method to solve it. I'm looking for some ideas here.

Answer 1

The Euler characteristic of a connected sum of surfaces $S_1$ and $S_2$ is given by $$\chi(S_1 \# S_2) = \chi(S_1) + \chi(S_2) - 2.$$ Recall that $\chi(K)=0$ fo the Klein bottle.

Reference: Euler characteristic of a connected sum of surfaces.