Benutzer:Hans G. Oberlack/H 0.1023974

The semicircle as base element.

General case

Segments in the general case

The radius of the semicircle $=r$

Perimeters in the general case

Perimeter of base semicircle$=2\cdot r+\pi \cdot r=(2+\pi )\cdot r$

Areas in the general case

Area of the base semicircle $={\frac {\pi \cdot r^{2}}{2}}$

Centroids in the general case

Centroid positions are measured from the lower left point of the surrounding rectangle with horizontal side length $2\cdot r$ and vertical side length r
Centroid positions of the base semicircle: $x_{c}=r\quad y_{c}={\frac {4\cdot r}{3\cdot \pi }}$

Normalised case

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

Segments in the normalised case

Segment of the base semicircle $r={\sqrt {\frac {2}{\pi }}}=0.79788...$

Perimeters in the normalised case

Perimeter of base semicircle$=(2+\pi )\cdot r=(2+\pi )\cdot {\sqrt {\frac {2}{\pi }}}=4.102397...$

Areas in the normalised case

Area of the base semicircle is by definition $=1$

Centroids in the normalised case

Centroid positions of the base semicircle: $x_{c}=r={\sqrt {\frac {2}{\pi }}}=0.79788...\quad y_{c}={\frac {4\cdot r}{3\cdot \pi }}={\frac {4}{3\cdot \pi }}\cdot {\sqrt {\frac {2}{\pi }}}={\sqrt {\frac {32}{9\cdot \pi ^{3}}}}=0.33863...$