When looking at regular polygons with all the diagonals filled in, I saw that concentric rings seem to form. Why does this occur? It's not so obvious with small $n$, but for larger $n$ it becomes increasingly clear.
To show what I mean, I have included the images of some regular polygons (the $n$ used as examples were chosen because I like them, but the pattern shows up for the other $n$s as well).
[Converting comment to answer, by request.]
In a circle of radius $r$, every chord of length $2s$ is tangent to the concentric circle of radius $\sqrt{r^2−s^2}$. Each such circle is the envelope of the associated infinite family of chords.
In a polygon, the diagonals of a particular length have a common tangent circle; for a polygon with lots of vertices, the families of diagonals are "visually dense enough" to make their circular envelopes apparent.