From the definition of the radian, its obvious that the arc length of the arc corresponding to the chord of such a triangle is merely theta in radians, but is there a formula of the length of this chord in terms of theta?
Yes. I think it's length=sqrt(2-2cos(theta)).
This formula certainly works for many of the values you mention (it worked for the ones I checked),
What values of theta produce chord lengths of 1/sqrt(2) and 1/(sqrt(3).
A chord length of 1/sqrt(2) is produced by theta=arcos(3/4), or roughly 20.7*
A chord length of 1/sqrt(3) is produced by theta=arcos(5/6) or roughly 16.8*
the chord lengths produced by thetas of 30, 72, 144, and 150.
If theta=30*, then chord length=sqrt(2-sqrt(3))
If theta=72*, then chord length=sqrt((5-sqrt(5))/2)
If theta=144*, then chord length=sqrt((5+sqrt(5))/2)
If theta=150*, then chord length=sqrt(2+sqrt(3))
I'm reasonably sure all the above is right. Someone else may have other ideas.
Since simply answering my questions would make for a very short discussion, fill free to post any other interesting tidbits about triangle, circles, or other geometric shapes.
i really like what happens to mobius strips when you cut them at various widths.