Can $\pi$ be a ratio of angles? I know that $\pi$ is the ratio between various measurements in 2 and 3 dimensional shapes. (For example, $V=\pi r^2h$ for a right circular cylinder can be written as $\pi=\frac{V}{r^2h}$) The golden ratio, on the other hand, is a ratio of angles also.
I suppose the answer is no, but is there are way of writing $\pi$ (or $\pi$ degrees) as a ratio that has angle measurements (that don't otherwise cancel out)?
To clarify, only measurements (whether angles, lines, volumes, etc.) that naturally occur in any shape, for example, an angle of a regular pentagon is always 72 degrees, and can be used. Directly using $\pi$ in the ratio is not allowed as an angle of $\pi$ degrees does not naturally occur in any shape (Pls. correct me if I'm wrong).

Let $OAB$ be a sector centered at $O$, with length of arc $AB$ equals to radius $OA=OB$. The sum of the two angles $\angle A$ and $\angle B$, each formed between radius and tangent, is $\pi$ times of $\angle O$.
