Hi Jeffrey

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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),

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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*

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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.

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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.