The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to be.
Inscribe inside a circle of radius $1$ a regular polygon with $n$-vertices, labeled consecutively. Draw a radius from the center to each vertex. For $i=1,\ldots,n$, place a bead $b_i$ on the radius connected to vertex $i$. We can slide each bead to different positions on its radius, so the state of the beads is described with a vector $(x_1,\ldots,x_n)\in [0,1]^n$, where $x_i$ is the distance of $b_i$ from the center. Let $P(x_1,\ldots,x_n)$ be the polygon formed by connecting each $b_i$ to its immediate neighbors with a straight line segment. Each pair of neighbors along with the center form a triangle, and summing the area of all such triangles gives the area of $P(x_1,\ldots,x_n)$:
$$A(x_1,\ldots,x_n)=\frac{1}{2}\sin\left(\frac{2\pi}{n}\right)\left(x_nx_1+\sum_{i=1}^{n-1}x_ix_{i+1}\right)$$
The total area of the regular $n$-gon is
$$A_n=\frac{n}{2}\sin\left(\frac{2\pi}{n}\right)$$
The ratio of the two areas defines a function:
$$f(x_1,\ldots,x_n)=\frac{1}{n}\left(x_nx_1+\sum_{i=1}^{n-1}x_ix_{i+1}\right)$$
If we choose the $x_i$ uniformly at random, what is the probability that $f(x_1,\ldots,x_n)>c$ for a given $c\in[0,1]$?
Since $f$ is continuous, the set $E_c=\{(x_1,\ldots,x_n)\mid f(x_1,\ldots,x_n)>c\}\subset [0,1]^n$ is measurable, so I'm just looking for $m(E_c)$, but I'm not too sure how to find it.