Let $A$ be a circle of radius $r$, and let $B$ be a circle of radius $2r-\epsilon$ for some small $\epsilon>0$, centered at a point on $A$.
How can one determine the outer arc of $A$ determined by where $B$ cuts $A$?
In the picture, I would like to determine the length of the outer green arc at the top of $A$.
I searched around and found similar questions, but none that matched what I am looking for.
A trigonometric approach:
Let's assume $X$ and $Y$ are the intersections of the circles $A$ and $B$, and $O$ and $Z$ are the centers of the circle $A$ and $B$, respectively. Consider $\triangle XOZ$ and $\triangle YOZ$, which are congruent isosceles triangles. Thus,
$$\angle XOZ =\angle YOZ =2 \arcsin \frac{r-\frac{\epsilon}{2}}{r} \implies \angle YOX =2\pi -4 \arcsin \frac{r-\frac{\epsilon}{2}}{r}.$$
Now, you can determine the length of the desired arc.