File:FS CRC dia.png

Base is the circle of given radius ${\displaystyle r_{0}}$ around point ${\displaystyle S_{0}}$
Inscribed is the largest possible right triangle with side length ${\displaystyle a_{1}}$.
Added is the largest circle ${\displaystyle r_{2}}$ around point ${\displaystyle S_{2}}$
.

0) The radius of the base circle ${\displaystyle =r_{0}}$
1) Side of the triangle ${\displaystyle a_{1}={\sqrt {2}}\cdot r_{0}\quad }$, see CR dia
2) The radius of the inscribed circle ${\displaystyle r_{2}={\frac {1}{2}}\cdot r_{0}}$

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}}$
1) Perimeter of the triangle ${\displaystyle P_{1}=({\sqrt {2}}+1)\cdot 2r_{0}}$
2) Perimeter of inscribed circle ${\displaystyle P_{2}=2\cdot \pi \cdot r_{2}=2\cdot \pi \cdot {\frac {1}{2}}\cdot r_{0}=\pi \cdot r_{0}}$

0) Area of the base circle ${\displaystyle A_{0}=\pi \cdot r_{0}^{2}}$
1) Area of the inscribed triangle ${\displaystyle A_{1}={\frac {a_{1}^{2}}{2}}={\frac {({\sqrt {2}}r_{0})^{2}}{2}}={\frac {2r_{0}^{2}}{2}}=r_{0}^{2}={\frac {A_{0}}{\pi }}}$
2) Area of the inscribed circle ${\displaystyle A_{2}=\pi \cdot r_{2}^{2}=\pi \cdot ({\frac {1}{2}}\cdot r_{0})^{2}={\frac {\pi }{4}}\cdot r_{0}^{2}={\frac {A_{0}}{4}}}$

Covered surface of the base shape ${\displaystyle C_{b}={\frac {A_{1}+A_{2}}{A_{0}}}={\frac {{\frac {A_{0}}{\pi }}+{\frac {A_{0}}{4}}}{A_{0}}}={\frac {4+\pi }{4\pi }}\quad \approx 56.8\%}$

0) Centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Centroid position of the inscribed triangle: ${\displaystyle S_{1}=-{\frac {1}{3{\sqrt {2}}}}\cdot (1+i)\cdot r_{0}\quad \approx -0.236\cdot (1+i)\cdot r_{0}\quad }$, see CR dia
2) Centroid position of the inscribed circle: ${\displaystyle S_{2}={\frac {r_{2}}{\sqrt {2}}}\cdot (1+i)={\frac {r_{0}}{2{\sqrt {2}}}}\cdot (1+i)\quad \approx 0.354(1+i)\cdot r_{0}}$

W) Weighted average centroid: ${\displaystyle C_{w}=\left({\frac {3\pi -8}{6{\sqrt {2}}(5\pi +4)}}\right)(1+i)\cdot r_{0}\quad \approx 0.0085\cdot (1+i)\cdot r_{0}\quad }$, see Calculation 1

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 ${\displaystyle k>0}$. This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Orientated centroid position of the inscribed triangle: ${\displaystyle S_{1}=\left(0-{\frac {1}{3}}\cdot i\right)\cdot r_{0}\quad \approx (0-0.333i)\cdot r_{0}\quad }$
2) Orientated centroid position of the inscribed circle: ${\displaystyle S_{2}=r_{2}\cdot (0+i)=\left(0+{\frac {1}{2}}\cdot i\right)\cdot r_{0}=(0+0.500i)\cdot r_{0}}$

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 triangle ${\displaystyle a_{1}={\sqrt {2}}\cdot {\frac {1}{\sqrt {\pi }}}={\sqrt {\frac {2}{\pi }}}\quad \approx 0.798}$
2) Radius of the inscribed circle ${\displaystyle r_{2}={\frac {1}{2{\sqrt {\pi }}}}\quad \approx 0.282}$

0) Perimeter of base circle ${\displaystyle P_{0}=2\cdot \pi \cdot r_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}\quad \approx 3.5449077}$
1) Perimeter of the inscribed triangle ${\displaystyle P_{1}=({\sqrt {2}}+1)\cdot 2\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx 2.7241483}$
2) Perimeter of inscribed circle: ${\displaystyle P_{2}=\pi \cdot {\frac {1}{\sqrt {\pi }}}={\sqrt {\pi }}\quad \approx 1.7724539}$
S) Sum of perimeters ${\displaystyle P_{s}=P_{0}+P_{1}+P_{2}\quad \approx 8.0415098}$

0) Area of the base circle ${\displaystyle A_{0}=1}$
1) Area of the inscribed triangle ${\displaystyle A_{1}=r_{0}^{2}=({\frac {1}{\sqrt {\pi }}})^{2}={\frac {1}{\pi }}\quad \approx 0.318}$
2) Area of the inscribed circle ${\displaystyle A_{2}={\frac {A_{0}}{4}}=0.250}$

0) Centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Centroid position of the inscribed triangle: ${\displaystyle S_{1}=-{\frac {1}{3{\sqrt {2\pi }}}}(1+i)\quad \approx -0.133\cdot (1+i)}$
2) Centroid position of the inscribed circle: ${\displaystyle S_{2}={\frac {1}{2{\sqrt {2}}}}\cdot {\frac {1}{\sqrt {\pi }}}\cdot (1+i)\quad \approx 0.199\cdot (1+i)}$

W) Weighted average centroid: ${\displaystyle C_{w}=\left({\frac {3\pi -8}{6{\sqrt {2}}(5\pi +4)}}\right)(1+i){\frac {1}{\sqrt {\pi }}}\quad \approx 0.0048\cdot (1+i)\quad }$

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 ${\displaystyle k>0}$. This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle: ${\displaystyle S_{0}=0+0i}$
1) Orientated centroid position of the inscribed triangle: ${\displaystyle S_{1}=\left(0-{\frac {1}{3}}\cdot i\right)\cdot {\frac {1}{\sqrt {\pi }}}=\left(0-{\frac {1}{3{\sqrt {\pi }}}}\cdot i\right)\quad \approx (0-0.188i)\quad }$
2) Orientated centroid position of the inscribed circle: ${\displaystyle S_{2}=r_{2}\cdot (0+i)=\left(0+{\frac {1}{2}}\cdot i\right)\cdot {\frac {1}{\sqrt {\pi }}}=\left(0+{\frac {1}{2{\sqrt {\pi }}}}\cdot i\right)\quad \approx (0+0.282i)}$

${\displaystyle C_{w}={\frac {C_{0}\cdot A_{0}+C_{1}\cdot A_{1}+C_{2}\cdot A_{2}}{A_{0}+A_{1}+A_{2}}}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=(0+0i)\cdot A_{0}+\left(-{\frac {1}{3{\sqrt {2}}}}\right)(1+i)\cdot r_{0}\cdot A_{1}+\left({\frac {1}{2{\sqrt {2}}}}\right)(1+i)\cdot r_{0}\cdot A_{2}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=\left(-{\frac {1}{3{\sqrt {2}}}}\right)(1+i)\cdot r_{0}\cdot r_{0}^{2}+\left({\frac {1}{2{\sqrt {2}}}}\right)(1+i)\cdot r_{0}\cdot {\frac {\pi }{4}}\cdot r_{0}^{2}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=\left(-{\frac {1}{3{\sqrt {2}}}}+{\frac {1}{2{\sqrt {2}}}}\cdot {\frac {\pi }{4}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=\left(-{\frac {1}{3{\sqrt {2}}}}+{\frac {\pi }{8{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=\left(-{\frac {8}{24{\sqrt {2}}}}+{\frac {3\pi }{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (A_{0}+A_{1}+A_{2})=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (\pi r_{0}^{2}+A_{1}+A_{2})=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (\pi r_{0}^{2}+r_{0}^{2}+A_{2})=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (\pi r_{0}^{2}+r_{0}^{2}+{\frac {\pi }{4}}\cdot r_{0}^{2})=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot (\pi +1+{\frac {\pi }{4}})\cdot r_{0}^{2}=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot \left({\frac {4\pi +4+\pi }{4}}\right)\cdot r_{0}^{2}=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}^{3}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot \left({\frac {5\pi +4}{4}}\right)=\left({\frac {3\pi -8}{24{\sqrt {2}}}}\right)(1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow C_{w}\cdot \left(5\pi +4\right)=\left({\frac {3\pi -8}{6{\sqrt {2}}}}\right)(1+i)\cdot r_{0}}$
${\displaystyle \quad \Leftrightarrow C_{w}=\left({\frac {3\pi -8}{6{\sqrt {2}}(5\pi +4)}}\right)(1+i)\cdot r_{0}\quad \approx 0.0085\cdot (1+i)\cdot r_{0}}$

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