I have not solved this but this is how I have been thinking.
I started with a circle of radius 1
The area is pi and the circumference is 2pi so
\(Ratio=\frac{A}{P}=\frac{1}{2}\)
I inscribed a regular polygon in the circle as shown. In this case, since it is a trangle, n will be 3, but the formulas are general
I worked out x using the cosine rule. and I worked out that if the triangle is to have an area of pi (like the circle) the dilation factor is as given.
So now the equivalaent ratio is \(R_2=\frac{\pi}{P}\)
My intentions was to show that P decreased as n increased hense R2 increases as n increases.
However it got to complicated and i couldn't make it work.
Maybe I piced the wrong values as my starting point idk
I did not use your apothem idea. Maybe that would have worked out better ....