Displays the largest equilateral hexagon within a circle

Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$
Inscribed is the largest possible equilateral triangle.

0) The radius of the base circle: ${\displaystyle r_{0}=|AS_{0}|=|BS_{0}|}$
1) Side length of the hexagon: ${\displaystyle a_{1}=|AB|=|AS_{0}|=r_{0}\quad }$, since the hexogon is equilateral

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}}$
1) Perimeter of the hexagon ${\displaystyle P_{1}=6\cdot a_{1}=6\cdot r_{0}}$
S) Sum of perimeters: ${\displaystyle P_{s}=P_{0}+P_{1}=(2\pi +6)\cdot r_{0}}$

0) Area of the base circle ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed hexagon ${\displaystyle A_{1}={\frac {3{\sqrt {3}}}{2}}\cdot r_{0}^{2}\quad }$

Covered surface of base shape: ${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {3{\sqrt {3}}}{2\pi }}\quad \approx 82.7\%}$

Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed hexagon: ${\displaystyle S_{1}=S_{0}=0+0i}$

In the normalised case the area of the base is set to 1.
${\displaystyle A_{0}=1\quad \Rightarrow \pi \cdot r_{0}^{2}=1\quad \Rightarrow r_{0}^{2}={\frac {1}{\pi }}\quad \Rightarrow r_{0}={\frac {1}{\sqrt {\pi }}}}$

0) Radius of the base circle ${\displaystyle r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564}$
1) Side length of the inscribed hexagon ${\displaystyle a_{1}=r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564}$

0) Perimeter of base square ${\displaystyle P_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}\quad \approx 3.5449077}$
1) Perimeter of the inscribed hexagon ${\displaystyle P_{1}=6\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx 3.3851375}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}=(2\pi +6)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx 6.9300452}$

0) Area of the base square ${\displaystyle A_{0}=1}$
1) Area of the inscribed hexagon ${\displaystyle A_{1}={\frac {3{\sqrt {3}}}{2}}\cdot \left({\frac {1}{\sqrt {\pi }}}\right)^{2}={\frac {3{\sqrt {3}}}{2\pi }}\quad \approx 0.827}$

Centroid positions are measured from the centroid point of the base shape.
0) Centroid positions of the base square: ${\displaystyle S_{0}=0+0i}$
1) Centroid positions of the inscribed hexagon:${\displaystyle S_{1}=0+0i}$

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