Take the following problem:

We break a stick of length one into five pieces by choosing four breaks along the stick randomly and independently and then breaking at those points. Assume that the pieces form a $5$-gon.

This occurs with probability $1-(5/16) = {11\over16}$ (this comes from the paper https://atlas.mat.ub.edu/personals/dandrea/emiliano_gomez.ps which states that an $n$-gon is formed from $n$ breaks with probability $1-{n\over2^{n-1}}$).

Using this distribution of lengths and assuming that a cyclic $5$-gon has been formed, what is the expected value of the shape's area?

Any ideas would help, thanks.

Suppose we break a stick of length one at four randomly and independently chosen points and that the resulting pieces form a pentagon.

Such a pentagon can be formed with probability $1-(5/16) = {11\over16}$ (see https://atlas.mat.ub.edu/personals/dandrea/emiliano_gomez.ps, which states that an $n$-gon is formed from $n-1$ breaks with probability $1-{n\over2^{n-1}}$).

Using this distribution of lengths and assuming that a cyclic pentagon has been formed, what is the expected value of the pentagon's area?