User:Hans G. Oberlack/Sandkiste

Elements

Base is the circle of given radius $r_{0}$ around point $S_{0}$
Inscribed is the largest possible fan (circular sector with $\angle 120^{\circ })$
.

General case

Segments in the general case

0) The radius of the base circle: $r_{0}$
1) The radius of the inscribed fan: $r_{1}={\frac {2}{\sqrt {3}}}\cdot r_{0}\quad \approx 1.155\cdot r_{0}\quad$, see Calculation 1

Perimeters in the general case

0) Perimeter of base circle $P_{0}=2\cdot \pi \cdot r_{0}\quad \approx 6.283\cdot r_{0}\quad$
1) Perimeter of the inscribed fan: $P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right)\cdot r_{0}\quad \approx 4.408\cdot r_{0}\quad$see calculation 2

Areas in the general case

0) Area of the base circle $A_{0}=\pi \cdot r_{0}^{2}$
1) Area of the inscribed fan: $A_{1}={\frac {4}{9}}\cdot \pi r_{0}^{2}={\frac {4}{9}}\cdot A_{0}\quad$, see Calculation 3

Centroids in the general case

The centroid positions of the following shapes will be expressed orientated so that the first shape n with $S_{n}\neq S_{0}$ will be of type $S_{n}=0-ki$ with $k>0$. The graphical representation does correspond to the mathematical expression.
0) Centroid position of the base circle: $S_{0}=0+0i=0$
1) Centroid position of the inscribed fan: $S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot r_{0}\quad \approx (0-0.059i)\cdot r_{0}$, see calculation 4

Normalised case

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

Segments in the normalised case

0) Radius of the base circle: $r_{0}={\frac {1}{\sqrt {\pi }}}\quad \approx 0.564$
1) Radius of the inscribed fan: $r_{1}={\frac {2}{\sqrt {3}}}\cdot {\frac {1}{\sqrt {\pi }}}={\frac {2}{\sqrt {3\pi }}}\quad \approx 0.651$

Perimeters in the normalised case

0) Perimeter of base square: $P_{0}=2\cdot \pi \cdot r_{0}=2\cdot \pi \cdot {\frac {1}{\sqrt {\pi }}}=2\cdot {\sqrt {\pi }}\quad \approx 3.544$
1) Perimeter of the fan: $P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right){\frac {1}{\sqrt {\pi }}}\quad \approx 2.485$

Areas in the normalised case

0) Area of the base circle: $A_{0}=1$
1) Area of the fan: $A_{1}={\frac {4}{9}}\cdot A_{0}={\frac {4}{9}}\quad \approx 0.444$

Covered surface of the base shape

$C_{b}={\frac {A_{1}}{A_{0}}}={\frac {4}{9}}\quad \approx 44\%$

Centroids in the normalised case

Centroid positions are measured from the centroid point of the base shape.
0) Centroid of the base circle: $S_{0}=0$
1) Centroid of the inscribed fan:$S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot {\frac {1}{\sqrt {\pi }}}\quad \approx (0-0.033i)$

Calculations

Equations of given elements and relations

(1) $|S_{0}A|=|S_{0}C|=|S_{0}B|=r_{0}$
(2) $|AD|=|CD|=r_{1}$
(3) $\delta =60^{\circ }={\frac {\pi }{3}}\quad$, since the angle in $D=120^{\circ }$
(4) $\sigma =90^{\circ }={\frac {\pi }{2}}\quad$, since the apothem ${\overline {DS_{0}}}$ is perpendicular to the chord ${\overline {AC}}$ .

Calculation 1

The radius $r_{1}$ is calculated:
${\frac {|AD|}{\sin \sigma }}={\frac {|AS_{0}|}{\sin \delta }}\quad$, applying the law of sines on the triangle $\triangle {ADS_{0}}$
$\quad \Leftrightarrow {\frac {r_{1}}{\sin \sigma }}={\frac {|AS_{0}|}{\sin \delta }}\quad$, applying equation (2)
$\quad \Leftrightarrow {\frac {r_{1}}{1}}={\frac {|AS_{0}|}{\sin \delta }}\quad$, applying equation (3)
$\quad \Leftrightarrow r_{1}={\frac {|AS_{0}|}{\sin \delta }}\quad$, eliminating 1 off the denominator
$\quad \Leftrightarrow r_{1}={\frac {r_{0}}{\sin \delta }}\quad$, applying equation (1)
$\quad \Leftrightarrow r_{1}={\frac {r_{0}}{{\frac {1}{2}}{\sqrt {3}}}}\quad$, applying the sin formula
$\quad \Leftrightarrow r_{1}={\frac {2\cdot r_{0}}{\sqrt {3}}}\quad \approx 1.155\cdot r_{0}$, rearranging

Calculation 2

Perimeter of the inscribed fan:
$\quad P_{1}=2r_{1}+{\frac {2\pi }{3}}\cdot r_{0}$
$\Leftrightarrow P_{1}=2\cdot \left({\frac {2}{\sqrt {3}}}\cdot r_{0}\right)+{\frac {2\pi }{3}}\cdot r_{0}\quad$, applying calculation 1
$\Leftrightarrow P_{1}=\left({\frac {4}{\sqrt {3}}}+{\frac {2\pi }{3}}\right)\cdot r_{0}\quad$, rearranging
$\Leftrightarrow P_{1}=\left({\frac {4{\sqrt {3}}}{3}}+{\frac {2\pi }{3}}\right)\cdot r_{0}\quad$, extending the fraction
$\Leftrightarrow P_{1}={\frac {2}{3}}\left(2{\sqrt {3}}+\pi \right)\cdot r_{0}\quad \approx 4.408\cdot r_{0}$, multiplying

Calculation 3

Area of the inscribed fan:
$\quad A_{1}={\frac {1}{3}}\cdot \pi r_{1}^{2}$
$\Leftrightarrow A_{1}={\frac {1}{3}}\cdot \pi \cdot \left({\frac {2\cdot r_{0}}{\sqrt {3}}}\right)^{2}\quad$, applying calculation 1
$\Leftrightarrow A_{1}={\frac {1}{3}}\cdot \left({\frac {2}{\sqrt {3}}}\right)^{2}\cdot \pi r_{0}^{2}\quad$, rearranging
$\Leftrightarrow A_{1}={\frac {1}{3}}\cdot {\frac {4}{3}}\cdot \pi r_{0}^{2}\quad$, squaring
$\Leftrightarrow A_{1}={\frac {4}{9}}\cdot \pi r_{0}^{2}={\frac {4}{9}}\cdot A_{0}\quad$, rearranging

Calculation 4

Centroid of the fan relative to the centroid of the circle.The centroid positions of the following shapes will be expressed orientated so that the first shape n with $S_{n}\neq S_{0}$ will be of type $S_{n}=0-ki$ with $k>0$. The graphical representation does correspond to the mathematical expression.

$\quad S_{1}=S_{0}+{\overrightarrow {S_{0}D}}+{\overrightarrow {DS_{1}}}\quad$
$\Leftrightarrow S_{1}=(0+0i)+{\overrightarrow {S_{0}D}}+{\overrightarrow {DS_{1}}}\quad$
$\Leftrightarrow S_{1}=(0+0i)+(0+|DS_{0}|\cdot i)-(0+|DS_{1}|\cdot i)\quad$
$\Leftrightarrow S_{1}=(0+|DS_{0}|\cdot i-|DS_{1}|\cdot i)\quad$, rearranging
$\Leftrightarrow S_{1}=(0+\left(|DS_{0}|-|DS_{1}|\right)\cdot i)\quad$, rearranging
$\Leftrightarrow S_{1}=(0+(r_{1}\cdot \sin {30^{\circ }}-|DS_{1}|)\cdot i)\quad$, since ${\frac {|DS_{0}|}{\sin {30^{\circ }}}}={\frac {r_{1}}{\sin \sigma }}=r_{1}$

$\Leftrightarrow S_{1}=(0+(r_{1}\cdot {\frac {1}{2}}-|DS_{1}|)\cdot i)\quad$, applying the sine formula
$\Leftrightarrow S_{1}=(0+\left(r_{1}\cdot {\frac {1}{2}}-{\frac {2r_{1}\sin \delta }{3\delta }}\right)\cdot i)\quad$, applying the centroid formula
$\Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {2\sin \delta }{3\delta }}\right)\cdot r_{1}\cdot i)\quad$, extracting
$\Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {2{\frac {\sqrt {3}}{2}}}{3\delta }}\right)\cdot r_{1}\cdot i)\quad$, applying the sine formula
$\Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{3\delta }}\right)\cdot r_{1}\cdot i)\quad$, shortening the fraction
$\Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{3{\frac {\pi }{3}}}}\right)\cdot r_{1}\cdot i)\quad$, applying equation 3
$\Leftrightarrow S_{1}=(0+\left({\frac {1}{2}}-{\frac {\sqrt {3}}{\pi }}\right)\cdot r_{1}\cdot i)\quad$, shortening the fraction
$\Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{2\pi }}\right)\cdot r_{1}\cdot i)\quad$, rearranging
$\Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{2\pi }}\right)\cdot {\frac {2\cdot r_{0}}{\sqrt {3}}}\cdot i)\quad$, applying calculation 1
$\Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot r_{0}\cdot i)\quad$, rearranging
$\Leftrightarrow S_{1}=(0+\left({\frac {\pi -2{\sqrt {3}}}{\pi {\sqrt {3}}}}\right)\cdot i)\cdot r_{0}\quad \approx (0-0.059i)\cdot r_{0}$