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).

If you mean the ratio of angles expressed in an integer or rational number of degrees then no, because $\pi$ is irrational.
Indeed, $\theta$ degrees is the same as $\frac{\pi\theta}{180}$. In a fraction of rational degrees, $\pi$ cancels and you're left with a rational number.
