Expected Triangle In Regular N-gon

Note that the general formula for the expected area of a triangle formed in this scenario is

$E(A_{N}) = \dfrac{3N*\cot(\frac{\pi}{N})}{2(N - 1)(N - 2)}$ .

Now, assuming that the radical can be attributed to the $\cot(\frac{\pi}{N})$ factor, we check to see what value of $N$ we get from the equation

$\dfrac{5}{24} = \dfrac{3N}{2(N - 1)(N - 2)}$

$\Longrightarrow 5*(N^{2} - 3N + 2) = 36*N$

$\Longrightarrow 5*N^{2} - 51*N + 10 = 0 \Longrightarrow (5N - 1)(N - 10) = 0 \Longrightarrow N = 10$ ,

as $N$ must be an integer. After confirming (using a calculator) that $\cot(\frac{\pi}{10}) = \sqrt{5 + 2\sqrt{5}}$ , we can conclude that $N = \boxed{10}$ .

Here is the paper where the general formula stated above is proved.

Each triangle has area is appropriate to: $\frac{2 \times \pi}{n} = \frac{5 \sqrt{5 + 2 \times \sqrt{5}}}{24} \Rightarrow n=\boxed{10}$

Place N-gon at the center of the Argand diagram and label each vertices $v_{1}, v_{2}, v_{3}, ..., v_{N}$ . ( $v_{1}$ at (1, 0) counterclockwise)

Given three vertices, the area of the triangle is equal to $\frac{1}{2}|(\vec{v_{a}} - \vec{v_{b}}) \times (\vec{v_{b}} - \vec{v_{c}})|$ where $\vec{v_{a}}, \vec{v_{b}}, \vec{v_{c}}$ are radius vectors.

Basically, the code below gets the mean (average) value of triangle areas for each possible value of N.

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from math import pi
from cmath import rect
from itertools import combinations

def vertex_angle(vertex_number, N):
    """Returns the angle (in radians) of nth unit radius vector
    with respect to +x axis"""
    return 2*pi*(vertex_number - 1)/N

def cross_product_magnitude(vector1, vector2):
    """Returns the magnitude of vector1 x vector2"""
    return abs((vector1 * vector2.conjugate()).imag)

for N in range(3, 15+1):
    sum_areas = 0.0
    num_triangles = N*(N-1)*(N-2)/6 # equal to NC3

    for vertices in combinations(range(1, N+1), 3):
        vectors = [rect(1, va) for va in [vertex_angle(vertex, N) for vertex in vertices]]
        area = 0.5*cross_product_magnitude(vectors[1] - vectors[0], vectors[2] - vectors[0])
        sum_areas += area

    print "(N, mean_area): ", N, sum_areas/num_triangles

Looking up the value of N where expected area is approximately equal to 0.641184, the answer is 10.