Expected Triangle In Regular N-gon

Calculus Level 3

Consider the \(N\) vertices of a regular \(N-\)gon that is inscribed in a unit circle. There are \( { N \choose 3 } \) possible subsets of 3 vertices, which form triangles. If the expected area of the triangle is

\[ \frac{ 5 \sqrt{ 5 + 2 \sqrt{5} } } { 24 } , \]

what is the value of \(N\)?

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