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.
I look at your diagram and see:
And
I solved those (using substitution, then the quadratic formula) and got
Whether or not a negative length is meaningful in this context is an exercise left to the reader
Giving (for L=5.44):
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.