I found a quiz on advanced trigonometry and geometry online and I was doing it for fun and it asks
Prove that for any regular polygon with side length $S$ and number of sides $N$ that $A = S N \cot(\pi/N)/4$
I have been stumped for 30 minutes now. Can somebody help?
HINT Here is one approach. Since it's a regular polygon, call its center $C$ and draw edges from $C$ to every vertex of the polygon. Note you end up with $N$ identical triangles, where the central angle is $2\pi/n$ and the length of the opposite side is $S$. You also know that all such triangles are isosceles by symmetry.
It suffices to prove the area of one such triangle is $S\cot(\pi/N)/4$, so now you can compute the length of the height from $C$ onto the side of length $S$ and compute the area. Can you finish it?