Benutzer:Hans G. Oberlack/V 0.0292122

The Quartercircle as base element.

General case

Segments in the general case

The radius of the quarter circle $=r$

Perimeters in the general case

Perimeter of base quarter circle$=2\cdot r+{\frac {\pi \cdot r}{2}}=(2+{\frac {\pi }{2}})\cdot r$

Areas in the general case

Area of the base quarter circle $={\frac {\pi \cdot r^{2}}{4}}$

Centroids in the general case

Centroid positions are measured from the lower left point of the surrounding square
Centroid positions of the base quarter circle: $x_{c}={\frac {4\cdot r}{3\cdot \pi }}\quad y_{c}={\frac {4\cdot r}{3\cdot \pi }}$

Normalised case

In the normalised case the area of the base quarter circle is set to 1.
So $A={\frac {\pi \cdot r^{2}}{4}}=1\quad \Rightarrow r^{2}={\frac {4}{\pi }}\quad \Rightarrow r={\sqrt {\frac {4}{\pi }}}={\frac {2}{\sqrt {\pi }}}$

Segments in the normalised case

Segment of the base quarter circle $r={\frac {2}{\sqrt {\pi }}}=1.128379...$

Perimeters in the normalised case

Perimeter of base quarter circle$=(2+{\frac {\pi }{2}})\cdot r=(2+{\frac {\pi }{2}})\cdot {\frac {2}{\sqrt {\pi }}}=4.029212...$

Areas in the normalised case

Area of the base quarter circle is by definition $=1$

Centroids in the normalised case

Centroid positions of the base quarter circle: $x_{c}={\frac {4\cdot r}{3\cdot \pi }}={\frac {4}{3\cdot \pi }}\cdot {\sqrt {\frac {4}{\pi }}}={\sqrt {\frac {64}{9\cdot \pi ^{3}}}}=0.479...\quad y_{c}=x_{c}=0.479...$