ratio of chord to its arc

r = pi * u / [360 * SIN(u / 2)]

Well, Holy Sin, Batman:
r = pi * u / [360 * SIN(u / 2 * pi / 180)]

I will seek forgiveness for my Sin by reciting the Hoooooly Rosary
twice, skipping no beads...and sweating in fear for good measure...

Got a few bucks to lend me, buddy? Need groceries...

will forgive you provided the Sin is of a rational angle when expressed in radians
 
but then what is wrong with the following reasoning?

as a central angle increases from 0 to 180 degrees (sorry, I think in degrees ), its chord and arc also increase. but the ratio between the arc and chord also smoothly increases from 1 to pi/2 and must pass through every value between 1 and pi /2. That includes rational irrational and transcendental numbers.

so where am I going wrong?

so I guess this is wrapped up. I conclude that (contrary to what I thoughtt when I posted the question)
that indeed there are infinitely many such chord/angle pairs.
much thanks to Bob Brown for the link to proof that the central angle of such chords can not be rational.

(I am having a bit of a hard time wrapping my head around the idea of the measure of an angle being irrational.)

but that being the case I further conclude that no such chord can be the side of a regular polygon.

I will have one final comment in next post.
 
No one doing serious mathematics has used degrees in a hundred years.

Well I didn't mean that I can't work with radians. I actually taught high school math. (algebra geometry and trig).
my classes were lively, the students were engaged and did well on state-wide tests.
but it was a long time ago. When I encounter my students we greet as old friends. Many are grandparents.

so I am familiar with radians which are wonderful for higher mathematics, but degrees are still the thing for everyday stuff.

When I posted the question, I was a little nervous that I was asking something basic that I had forgotten.

turned out not to be the case since I was not bombarded by multiple (or any) posts pointing this out.
 
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