Expected Triangle In Regular N-gon

Calculus Level 2

Consider the NN vertices of a regular NN-gon that is inscribed in a unit circle. There are (N3) { N \choose 3 } possible subsets of 3 vertices, which form triangles. If the expected area of the triangle is

55+2524, \frac{ 5 \sqrt{ 5 + 2 \sqrt{5} } } { 24 } ,

what is the value of NN?

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