Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$
Inscribed is the largest possible fan (circular sector with ${\displaystyle \angle 120^{\circ })}$
.

0) The radius of the base circle: ${\displaystyle r_{0}}$
1) The radius of the inscribed fan: ${\displaystyle r_{1}={\frac {2}{\sqrt {3}}}\cdot r_{0}\quad \approx 1.155\cdot r_{0}\quad }$, see Calculation 1

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}\quad \approx 6.283\cdot r_{0}\quad }$
1) Perimeter of the inscribed fan: ${\displaystyle P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right)\cdot r_{0}\quad \approx 4.408\cdot r_{0}\quad }$see calculation 2

0) Area of the base circle ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed fan: ${\displaystyle A_{1}={\frac {4}{9}}\cdot \pi r_{0}^{2}={\frac {4}{9}}\cdot A_{0}\quad }$, see Calculation 3

The centroid positions of the following shapes will be expressed orientated so that the first shape n with ${\displaystyle S_{n}\neq S_{0}}$ will be of type ${\displaystyle S_{n}=0-ki}$ with ${\displaystyle k>0}$. The graphical representation does correspond to the mathematical expression.
0) Centroid position of the base circle: ${\displaystyle S_{0}=0+0i=0}$
1) Centroid position of the inscribed fan: ${\displaystyle S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot r_{0}\quad \approx (0-0.059i)\cdot r_{0}}$, see calculation 4

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) Radius of the inscribed fan: ${\displaystyle r_{1}={\frac {2}{\sqrt {3}}}\cdot {\frac {1}{\sqrt {\pi }}}={\frac {2}{\sqrt {3\pi }}}\quad \approx 0.651}$

0) Perimeter of base square: ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}\quad \approx 3.544}$
1) Perimeter of the fan: ${\displaystyle P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right){\frac {1}{\sqrt {\pi }}}\quad \approx 2.485}$

0) Area of the base circle: ${\displaystyle A_{0}=1}$
1) Area of the fan: ${\displaystyle A_{1}={\frac {4}{9}}\cdot A_{0}={\frac {4}{9}}\quad \approx 0.444}$

${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {4}{9}}\quad \approx 44\%}$

Centroid positions are measured from the centroid point of the base shape.
0) Centroid of the base circle: ${\displaystyle S_{0}=0}$
1) Centroid of the inscribed fan:${\displaystyle S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.033i)}$

(1) ${\displaystyle |S_{0}A|=|S_{0}C|=|S_{0}B|=r_{0}}$
(2) ${\displaystyle |AD|=|CD|=r_{1}}$
(3) ${\displaystyle \delta =60^{\circ }={\frac {\pi }{3}}\quad }$, since the angle in ${\displaystyle D=120^{\circ }}$
(4) ${\displaystyle \sigma =90^{\circ }={\frac {\pi }{2}}\quad }$, since the apothem ${\displaystyle {\overline {DS_{0}}}}$ is perpendicular to the chord ${\displaystyle {\overline {AC}}}$ .

The radius ${\displaystyle r_{1}}$ is calculated:
${\displaystyle {\frac {|AD|}{\sin \sigma }}={\frac {|AS_{0}|}{\sin \delta }}\quad }$, applying the law of sines on the triangle ${\displaystyle \triangle {ADS_{0}}}$
${\displaystyle \quad \Leftrightarrow {\frac {r_{1}}{\sin \sigma }}={\frac {|AS_{0}|}{\sin \delta }}\quad }$, applying equation (2)
${\displaystyle \quad \Leftrightarrow {\frac {r_{1}}{1}}={\frac {|AS_{0}|}{\sin \delta }}\quad }$, applying equation (3)
${\displaystyle \quad \Leftrightarrow r_{1}={\frac {|AS_{0}|}{\sin \delta }}\quad }$, eliminating 1 off the denominator
${\displaystyle \quad \Leftrightarrow r_{1}={\frac {r_{0}}{\sin \delta }}\quad }$, applying equation (1)
${\displaystyle \quad \Leftrightarrow r_{1}={\frac {r_{0}}{{\frac {1}{2}}{\sqrt {3}}}}\quad }$, applying the sin formula
${\displaystyle \quad \Leftrightarrow r_{1}={\frac {2\cdot r_{0}}{\sqrt {3}}}\quad \approx 1.155\cdot r_{0}}$, rearranging

Perimeter of the inscribed fan:
${\displaystyle \quad P_{1}=2r_{1}+{\frac {2\pi }{3}}\cdot r_{0}}$
${\displaystyle \Leftrightarrow P_{1}=2\cdot \left({\frac {2}{\sqrt {3}}}\cdot r_{0}\right)+{\frac {2\pi }{3}}\cdot r_{0}\quad }$, applying calculation 1
${\displaystyle \Leftrightarrow P_{1}=\left({\frac {4}{\sqrt {3}}}+{\frac {2\pi }{3}}\right)\cdot r_{0}\quad }$, rearranging
${\displaystyle \Leftrightarrow P_{1}=\left({\frac {4{\sqrt {3}}}{3}}+{\frac {2\pi }{3}}\right)\cdot r_{0}\quad }$, extending the fraction
${\displaystyle \Leftrightarrow P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right)\cdot r_{0}\quad \approx 4.408\cdot r_{0}}$, multiplying

Area of the inscribed fan:
${\displaystyle \quad A_{1}={\frac {1}{3}}\cdot \pi r_{1}^{2}}$
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{3}}\cdot \pi \cdot \left({\frac {2\cdot r_{0}}{\sqrt {3}}}\right)^{2}\quad }$, applying calculation 1
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{3}}\cdot \left({\frac {2}{\sqrt {3}}}\right)^{2}\cdot \pi r_{0}^{2}\quad }$, rearranging
${\displaystyle \Leftrightarrow A_{1}={\frac {1}{3}}\cdot {\frac {4}{3}}\cdot \pi r_{0}^{2}\quad }$, squaring
${\displaystyle \Leftrightarrow A_{1}={\frac {4}{9}}\cdot \pi r_{0}^{2}={\frac {4}{9}}\cdot A_{0}\quad }$, rearranging

Centroid of the fan relative to the centroid of the circle.The centroid positions of the following shapes will be expressed orientated so that the first shape n with ${\displaystyle S_{n}\neq S_{0}}$ will be of type ${\displaystyle S_{n}=0-ki}$ with ${\displaystyle k>0}$. The graphical representation does correspond to the mathematical expression.

${\displaystyle \quad S_{1}=S_{0}+{\overrightarrow {S_{0}D}}+{\overrightarrow {DS_{1}}}\quad }$
${\displaystyle \Leftrightarrow S_{1}=(0+0i)+{\overrightarrow {S_{0}D}}+{\overrightarrow {DS_{1}}}\quad }$
${\displaystyle \Leftrightarrow S_{1}=(0+0i)+(0+|DS_{0}|\cdot i)-(0+|DS_{1}|\cdot i)\quad }$
${\displaystyle \Leftrightarrow S_{1}=(0+|DS_{0}|\cdot i-|DS_{1}|\cdot i)\quad }$, rearranging
${\displaystyle \Leftrightarrow S_{1}=(0+\left(|DS_{0}|-|DS_{1}|\right)\cdot i)\quad }$, rearranging
${\displaystyle \Leftrightarrow S_{1}=(0+(r_{1}\cdot \sin {30^{\circ }}-|DS_{1}|)\cdot i)\quad }$, since ${\displaystyle {\frac {|DS_{0}|}{\sin {30^{\circ }}}}={\frac {r_{1}}{\sin \sigma }}=r_{1}}$

${\displaystyle \Leftrightarrow S_{1}=(0+(r_{1}\cdot {\frac {1}{2}}-|DS_{1}|)\cdot i)\quad }$, applying the sine formula
${\displaystyle \Leftrightarrow S_{1}=(0+\left(r_{1}\cdot {\frac {1}{2}}-{\frac {2r_{1}\sin \delta }{3\delta }}\right)\cdot i)\quad }$, applying the centroid formula
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {2\sin \delta }{3\delta }}\right)\cdot r_{1}\cdot i)\quad }$, extracting
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {2{\frac {\sqrt {3}}{2}}}{3\delta }}\right)\cdot r_{1}\cdot i)\quad }$, applying the sine formula
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{3\delta }}\right)\cdot r_{1}\cdot i)\quad }$, shortening the fraction
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{3{\frac {\pi }{3}}}}\right)\cdot r_{1}\cdot i)\quad }$, applying equation 3
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{\pi }}\right)\cdot r_{1}\cdot i)\quad }$, shortening the fraction
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{2\pi }}\right)\cdot r_{1}\cdot i)\quad }$, rearranging
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{2\pi }}\right)\cdot {\frac {2\cdot r_{0}}{\sqrt {3}}}\cdot i)\quad }$, applying calculation 1
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot r_{0}\cdot i)\quad }$, rearranging
${\displaystyle \Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot r_{0}\quad \approx (0-0.059i)\cdot r_{0}}$

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