What are Euler Characteristics of the following shapes in $\mathbb R^2 ?$
Circles (2,3,4) have two arc radii $(R>r).$ Difficulty was in finding its value for tangential contact case 3) using Gauss-Bonnet thm so that by changing center distance $c$ we can arrive at case 2 or case 4 as a function of center distance $c$.
The tangential case seems to evaluate to $0$ by Euler formula and to $1$ with Gauss-Bonnet thm. So please help in correct calculation.
