Best Answer bonanova, 29 April 2014 - 11:38 AM

Since I inspired this, I guess I should solve it.

**-gon has no straight portions.**

*n*It is rather a sequence of

**parabolic sections centered on the apothems (perpendicular lines to the sides.)**

*n*Sketch an

*-gon with center at (0,1) and bottom side on the line*

**n***y*= -1.

The origin lies at the midpoint of the apothem drawn to the bottom side.

The origin is thus one of the equidistant points.

The complete locus of points equidistant from the center and the bottom side is

*y*=

*x*

^{2}/4.

If we do this for each side, we join

**parabolic segments, and the area of their combined interior**

*n*is the area of the * n*-gon that is closer to the center than to any side.

Lines drawn from the center to the

*vertices define the points where the parabolic segments join.*

**n**From the center, draw a line downward at an angle pi/

*from the vertical.*

**n**Call the slope of that line

*m*where

*m*= 1/tan(pi/

*).*

**n**That line intersects the parabola at *x _{n}* = 2(-

*m*+ sqrt(

*m*

^{2}+1)),

*y*=

_{n}*x*

_{n}^{2}/4

Look at the rectangle defined by

*x*= 0,

*x*and

_{n}*y*= 0, 1 - its area is just A

_{r}=

*x*, comprising three parts

_{n}A triangular portion above the diagonal line with area A

_{t }= 1/2

*x*(1-

_{n}*y*)

_{n}A lower area under the parabola with area A

_{p}= (1/12)

*x*

_{n}^{3}

The middle area comprises the points closer to the center than to any side.

The total area of interest is then 2

*(A*

**n**_{r}- A

_{t}- A

_{p}) =

**n***x*(1 + (1/12)

_{n}*x*

_{n}^{2})

Finally we find the total area of a regular

*-gon with apothem = 2.*

**n**From standard geometry, side = 2 apothem tan(pi/

*)*

**n**perimeter =

*x side*

**n**Area

_{n}= 1/2 apothem perimeter = 4

*tan(pi/*

**n***)*

**n**Note that

*tan(pi/*

**n***) goes to pi as*

**n***goes to infinity so Area*

**n**_{n=inf}= Area

_{circle}= 4pi.

Because the apothem becomes the radius for a circle.

The fraction of area of a regular

*-gon closer to its center than to any side is thus.*

**n***f*=

_{n}*x*(1+1/12

_{n}*x*

_{n}^{2}) / 4 tan(pi/

*) -> 1/4 as*

**n***-> infinity (circle).*

**n**Results for various values of

**n**n fraction

------+------------

3 0.1851851852

4 0.2189514165

5 0.2314757303

6 0.237604307

8 0.2432751885

10 0.2457666208

12 0.2470862357

20 0.2489644765

48 0.24982129

100000 0.2499999961