Can $\pi$ be a ratio of angles?

Question

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).

Answer 1

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 $\angle O$.

Answer 2

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.

Answer 3

How about a triangle with area $\pi$?

Start with the Area of a right triangle which is half the base times height, $A = \frac {bh}{2}$.

Now, with height $h = r = 1$, a base $b = C = \tau r = \tau$, and if we define $\pi := \tau/2$, we have:

$A = \frac {bh}{2} = \frac{\tau}{2} = \pi$

This is the triangle that Archimedes used. http://en.wikipedia.org/wiki/Area_of_a_disk#History I will let you think about the other two angles.