User:Hans G. Oberlack/Sandkiste: Difference between revisions

The semicircle as base element with given radius ${\displaystyle r_{0}}$
. Inscribed is the largest right triangle. Within this triangle the largest circle is inscribed

0) Radius of the semicircle: ${\displaystyle r_{0}}$
1) Side length of the right triangle: ${\displaystyle a_{1}={\sqrt {2}}\cdot r_{0}\quad \approx 1.414\cdot r_{0}\quad }$, see Calculation 1
2) Radius of the inscribed circle: ${\displaystyle r_{2}=({\sqrt {2}}-1)\cdot r_{0}\quad \approx 0.414\cdot r_{0}\quad }$, see Calculation 3

0) Perimeter of base semicircle: ${\displaystyle P_{0}=2\cdot r_{0}+\pi \cdot r_{0}=(2+\pi )\cdot r_{0}\quad \approx 5.142\cdot r_{0}}$
1) Perimeter of inscribed right triangle: ${\displaystyle P_{1}=2\cdot a_{1}+2\cdot r_{0}=2\cdot {\sqrt {2}}\cdot r_{0}+2\cdot r_{0}=2\cdot ({\sqrt {2}}+1)\cdot r_{0}\quad \approx 4.828\cdot r_{0}}$
2) Perimeter of the inscribed circle: ${\displaystyle P_{2}=2\pi \cdot r_{2}=2\pi \cdot ({\sqrt {2}}-1)\cdot r_{0}\quad \approx 2.603\cdot r_{0}\quad }$
S) Sum of perimeters: ${\displaystyle P_{s}=(4+(3+2{\sqrt {2}})\pi +2{\sqrt {2}})\cdot r_{0}\quad \approx 12.573\cdot r_{0}}$

0) Area of the base semicircle ${\displaystyle A_{0}={\frac {\pi }{2}}\cdot r_{0}^{2}\quad \approx 1.57\cdot r_{0}^{2}}$
1) Area of the inscribed right triangle ${\displaystyle A_{1}={\frac {a_{1}^{2}}{2}}={\frac {({\sqrt {2}}\cdot r_{0})^{2}}{2}}=r_{0}^{2}\quad }$
2) Area of the inscribed circle Failed to parse (unknown function "\cdor"): {\displaystyle A_2=\pi \cdot r_2^2 =\pi \cdot (\sqrt2-1)^2 \cdot r_0^2 =\pi \cdot (3-2\sqrt2) \cdot r_0^2 \quad \approx 0.539 \cdor r_0^2 \quad}

${\displaystyle C_{b}={\frac {A_{1}}{A_{0}}}={\frac {2}{\pi }}\quad \approx 0.637\approx 64\%}$