•  m0darn   ( @m0darn@lemmy.ca ) 
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      2 months ago

      I look at your diagram and see:

      ϴ= L/(L+R)
      

      And

      2π-ϴ = L/R
      

      I solved those (using substitution, then the quadratic formula) and got

      L= π-1 ± √(1+π²) ~= 5.44 or -1.16
      

      Whether or not a negative length is meaningful in this context is an exercise left to the reader

      Giving (for L=5.44):

      ϴ~= 0.845 ~~48.4° 
      

      I’m surprised that it solved to a single number, maybe I made a mistake.

      • That lines up pretty similarly with what I found also. The angle should be a constant since there is only one angle where the relationship would be true. I just left it in terms of π because I try to avoid rounding.

        Having said that, L would be a ratio of r; which I think lines up with what you found as well.