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:
[![Pent.gif][1]][1]
Edge lengths: $1, 3, 4, 2, 6$. Final circle radius $\approx 3.045$.