Limit with Average of area. Problem:
There is unit circle C on Coordinate plane.
For $n \in \mathbb{N}, n \geq 3,$ there are n-points on C at same intervals.
Define $S_n \colon$ Average of all triangle's areas which could be made by using n-points.
Compute $$\lim_{n\to\infty}S_n$$ 

I tried this for $n=3, n=4$ but this is little bit difficult when $n\geq5$.
I couldn't find any regulation, with using regular polygon on unit circle C.
How should I approach this problem?
 A: As you ask just for how to approach the problem, I'll show you how to determine each triangle area based on the subtending angles. With this, you can derive a formula for the average of the triangle areas for any given $n \ge 3$, and then try to determine the limit as $n \to \infty$. Consider the diagram below of one of the triangles inscribed in a unit circle:

$A, B$ and $C$ are the points on the circle, with $\alpha, \beta$ and $\gamma$ being the $3$ internal angles. The point where the $3$ internal lines meet is the center of the circle $O$ (not shown in the diagram). First consider just $\Delta AOC$. Since $\overline{AO} = \overline{CO} = 1$, it's an isosceles triangle, so you can draw the bisector of $O$ to $AC$ to have it meet at $D$, with it being the center of $AC$, as shown below:

Note that $\overline{AD} = \overline{CD} = \left|\sin\left({\frac{\gamma}{2}}\right)\right|$ and $\overline{OD} = \left|\cos\left({\frac{\gamma}{2}}\right)\right|$. Using the area of a triangle being one half the base times the height gives that $A(\Delta OAC) = \left|\frac{1}{2}\left(2\sin\left({\frac{\gamma}{2}}\right)\cos\left({\frac{\gamma}{2}}\right)\right)\right| = \left|\frac{\sin(\gamma)}{2}\right|$, where the Double angle formula for $\sin$ was used. The reason I use absolute values is because the calculated value may be negative if $\gamma \gt \pi$. In this case, you would need to subtract the triangle area instead. As $\sin(\gamma) \lt 0$ in that case, you can get the appropriate area when adding the triangle areas by subtracting the calculated area instead. Thus, in either case, you want to add $\frac{\sin(\gamma)}{2}$.
Doing the same calculations for the other $2$ smaller triangles and adding them gives $A(\Delta ABC) = \frac{\sin(\alpha) + \sin(\beta) + \sin(\gamma)}{2}$. For any given $n$, you can determine the appropriate sets of values of $\alpha, \beta$ and $\gamma$ for the inscribed triangles and then determine the average area. Do you think you can proceed from here to do this and then derive the limit of the triangle areas' average as $n \to \infty$?
