Comparing between regular polygons

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)

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)