Inscribing Regular Polygons in Circles

Geometry Level 5

Let \(n \ge 3\). A \(n\)-sided regular polygon is inscribed in a circle of radius \(1\). Then, all the diagonals of the polygon are drawn as well. Thus every vertex of the polygon is connected by a line segment to every other vertex.

Let \(L_n\) be the total length of line segments that are drawn by this process (including the perimeter). To the nearest thousandth, find the value of\(\displaystyle\lim_{n\to\infty}\frac{L_n}{n^2}\).

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