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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