File:FS F dia.png

The 120°-degree fan as base element.

0) The radius of the fan: ${\displaystyle r_{0}}$

0) Perimeter of base shape: ${\displaystyle P_{0}=2\cdot r_{0}+{\frac {2\pi \cdot r_{0}}{3}}=(2+{\frac {2\pi }{3}})\cdot r_{0}\quad \approx 4.094\cdot r_{0}}$

0) Area of the base shape ${\displaystyle A_{0}={\frac {\pi \cdot r_{0}^{2}}{3}}\approx 1.047\cdot r_{0}^{2}}$

0) By definition the centroid point of a base shape is ${\displaystyle S_{0}=0+0i}$

In the normalised case the area of the base shape is set to 1.
So ${\displaystyle A_{0}={\frac {\pi \cdot r_{0}^{2}}{3}}=1\quad \Rightarrow r_{0}^{2}={\frac {3}{\pi }}\quad \Rightarrow r_{0}={\sqrt {\frac {3}{\pi }}}}$

0) Radius of the base shape: ${\displaystyle r_{0}={\sqrt {\frac {3}{\pi }}}\quad \approx 0.977}$

0) Perimeter of base shape: ${\displaystyle P_{0}=(2+{\frac {2\pi }{3}})\cdot {\sqrt {\frac {3}{\pi }}}\approx 4.0010635}$

0) Area of the base shape is by definition ${\displaystyle A_{0}=1}$

0) Centroid of the base shape: ${\displaystyle S_{0}=0+0i}$

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