Not an answer,
just an illustration of the uniqueness as mentioned in Fedor Petrov's comment.
Start with a large radius $r$, and inscribe the segments along the arc of the circle.
Then shrink $r$ until the ends meet:
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[![Pent.gif][1]][1]
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<sup>
Edge lengths: $1, 3, 4, 2, 6$. Final circle radius $\approx 3.045$.
</sup>
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  [1]: https://i.sstatic.net/OVqtL.gif