If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle's radius, is there a formula by which I can calculate the length of the sagitta?

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If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle&#039;s radius, is there a formula by which I can calculate the length of the sagitta?

minnegodks · Accepted Answer

I hadn&#039;t even known the term before you made me look it up.:) I don&#039;t see a way to compute it explicitly, but it is in fact uniquely determined. This is because knowing   r  θ  =  A and   r  sin  ⁡  θ  =  B, there is a unique   θ(  0  &amp;lt;  θ  &amp;lt;  π      /    2) with                     sin        ⁡        θ            θ        =            B      A      , and then   r  =            A      θ       is unique as well. The sagitta is, of course,   r  (  1  −  cos  ⁡  θ  )  =  r  −            r      2        −          B      2      .Clarifying Comment: Here   A is half the arclength and   B is half the chord length.

If I know the length of a chord of a

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