In this experiment, I am adding the inradius (let's call it $A$) and circumradius (let's call it $B$) of different polygons with equal sides each equal $1$ (starting with a square and adding one side each time). The result is $A+B=C$ when side of polygon = 1.
When comparing the $C$ of one polygon with the $C$ of a polygon with one side more, the difference seems to go smaller, as if approaching a version of $\pi$ number with $0.$ before (possibly such as 0.314159265359…).
Can anyone confirm it or elaborate on it?
I can not go over a polygon with 1000 sides in my computation power, and would like to know what to expect while going towards a polygon with infinity sides.
Here are some examples:
4 sided polygon: $0.5 + 0.707106781 = 1.207106781$
5 sided polygon: $0.68819096 + 0.850650808 = 1.5388417680000002$ (Difference of 0.33173498700000015 from previous result)
6 sided polygon: $0.866025404 +1 = 1.866025404$$ (Difference of 0.3271836359999998 from previous result)
7 sided polygon: $1.0382607 + 1.15238244 = 2.1906431399999997$ (Difference of 0.32461773599999977 from previous result)
8 sided polygon: $1.20710678 + 1.30656296 = 2.51366974$ (Difference of 0.3230266000000004 from previous result)
9 sided polygon: $1.37373871 + 1.4619022 = 2.83564091$ (Difference of 0.32197116999999986 from previous result)
10 sided polygon: $1.53884177 + 1.61803399 = 3.15687576$ (Difference of 0.3212348500000002 from previous result)
11 sided polygon: $1.70284362+ 1.77473277 = 3.47757639$ (Difference of 0.3207006299999997 from previous result)
12 sided polygon: $1.8660254+ 1.93185165 = 3.79787705$ (Difference of 0.32030066 from previous result)
13 sided polygon: $2.02857974+ 2.08929073 = 4.11787047$ (Difference of 0.31999341999999986 from previous result)
14 sided polygon: $2.19064313 + 2.2469796 = 4.43762273$ (Difference of 0.31975226000000045 from previous result)
15 sided polygon: $2.35231505+ 2.40486717 = 4.757182220000001$ (Difference of 0.3195594899999996 from previous result)
…
999 sided polygon: $158.995264 + 158.99605 = 317.991314$
1000 sided polygon: $159.154419 + 159.155205 = 318.309624$ (Difference of 0.31830999999999676 from previous result)
Best Answer
Consider that the apothem $a$, circumcircle radius $c$, and edge of the polygon form a right triangle.
That is, we have two legs, one of length $1/2$, the other of length $a$, and a hypotenuse of length $c$. Then we have:
$$a = \frac{\tan\left(\frac{(n-2)\pi}{2n}\right)}{2}$$
$$c = \frac{1}{2}\csc\left(\frac{\pi}{n}\right)$$
Since the angle $\angle ac$ is always $\pi/n$.
We may simplify $a + c = \frac{1}{2}\cot\left(\frac{\pi}{2n}\right)$. Then what you seek to compute is:
$$\lim_{n \rightarrow \infty} \frac{1}{2}\left(\cot\left(\frac{\pi}{2n}\right) - \cot\left(\frac{\pi}{2n-2}\right)\right)$$
This in fact converges to $1/\pi$, which is $\approx 0.318$.