Hi, I've been trying to prove that the circle is the shape (that is regular, as regular shapes have the greatest area for the same n-gon) with the greatest area for a given perimeter. After seeing that the perimeters cancel out, I've deduced that it boils down to the length between apothems of the regular polygons versus the radius of the circle. I've attempted to show that the circle has the "longest apothem" (its radius) for a given perimeter, but I'm struggling. Thus, I've come here to ask for help. May someone aid me in this task?
+118687
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 ....
