Is it possible to have a geometry, in 2d, in which, for all circles of a certain radius $r_y$, $c=\frac{r}{2\pi}$?

Question

I was wondering if it is possible to have a geometry, in 2d, in which, for all circles with a certain radius $$r_y$$ but only for circles with radius $$r_y$$ $$c=\frac{r}{2\pi}$$ with $c$ being the circumference of the circle, and $r$ being the radius of the circle.

Answer 1

On a sphere, every circle with the fixed radius $r_y$ has the same circumference, and by choosing a circle's radius appropriately you can make the ratio of the circle's circumference to its radius be anything between $0$ and $2\pi.$ In particular, you can fix $r$ so that $c = \frac{r}{2\pi} < 2\pi r.$