Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from [geogebra applet][1]).  

[![enter image description here][2]][2]

I've been trying to find, in terms of $n$, bounds on the area of the largest cell, excluding the centre cell when $n$ is odd. 

It seems that, for every value of $n$, the largest non-centre cell has roughly similar area as the outermost cell, whose area is $\frac{1}{4}\tan{\frac{\pi}{n}}\approx\frac{\pi}{4n}$.

[![enter image description here][3]][3]

That is, $(\text{area of largest non-centre cell})$ is on the order of $n^{-1}$.

(This claim has numerical evidence: the first graph in [this answer][4] shows that, in a regular $n$-gon of *radius* $1$ with diagonals, up to $n=200$, the radius of the largest disk that fits within a cell, is approximately $3n^{-1.5}$; scaling to a regular $n$-gon of *side length* $1$, the largest disk would have an area of approximately $\frac{9}{4\pi n}$.)

Question:

>In a regular $n$-gon of side length $1$ with diagonals, what is the infimum and supremum of $n\times(\text{area of largest non-centre cell})$ ?

(The number of cells is [approximately $\frac{1}{24}n^4$][5] for large $n$.)

**EDIT:** If we don't know the answer to this question, then can we at least show that $n\times(\text{area of largest non-centre cell})$ has an upper bound?


[Cross-posted][6] on MSE.


  [1]: https://www.geogebra.org/m/AXE5mRgH
  [2]: https://i.sstatic.net/3JGmN.png
  [3]: https://i.sstatic.net/GmJmX.png
  [4]: https://math.stackexchange.com/a/4591233/398708
  [5]: https://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html
  [6]: https://math.stackexchange.com/questions/4593337/regular-polygon-with-diagonals-bounds-on-area-of-largest-cell?noredirect=1#comment9690605_4593337