User:Lilyuserin/SVG/zehneck

Allgemeines

Viele Aussagen über das regelmäßige Fünfeck gelten auch für das Zehneck und das regelmäßige Zwanzigeck. Der Goldene Schnitt $\varphi =$ ${\frac {1}{2}}({\sqrt {5}}+1)$ und die Zahl ${\sqrt {5}}$ spielen eine wichtige Rolle.

Anm.: die externen Links führen meist zu Wolfram Alpha mit der Eingabe der entsprechenden Formel.

Eckpunkte

Die $n$ Eckpunkte eines regelmäßigen N-Ecks mit Umkreisradius $R$ ergeben sich bekanntlich als

P_{k}=\left(R\cos {\frac {k\pi }{n}},R\sin {\frac {k\pi }{n}}\right)\quad k=0,\ldots ,n-1

Zunächst definiere ich $\varphi =$ ${\frac {1}{2}}({\sqrt {5}}+1)$ und $\Xi ={\sqrt {1-{\frac {\varphi ^{2}}{4}}}}={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$

so gilt $\sin 36^{o}$ $=\Xi ={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}},$ $\cos 36^{o}={\frac {\varphi }{2}}$ $\Rightarrow$ $\sin 72^{o}=\Xi *\varphi ={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}},$ $\cos 72^{o}={\frac {1}{2\varphi }}$

Eckpunkte tabellarisch

Winkel	$cos$	$sin$	Eckpunkt	R=754
$0^{o}$	$1$	$0$	$(1,0)$	(754,0)
$36^{o}={\frac {\pi }{5}}$	${\frac {\varphi }{2}}$	$\Xi$	$\left({\frac {1}{4}}(1+{\sqrt {5}}),{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(610,443.19)
$72^{o}={\frac {2\pi }{5}}$	${\frac {1}{2\varphi }}$	$\Xi *\varphi$	$\left({\frac {1}{4}}({\sqrt {5}}-1),{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(233,717.1)
$108^{o}={\frac {3\pi }{5}}$	$-{\frac {1}{2\varphi }}$	$\Xi *\varphi$	$\left(-{\frac {1}{4}}({\sqrt {5}}-1),{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(-233,717.1)
$144^{o}={\frac {4\pi }{5}}$	$-{\frac {\varphi }{2}}$	$\Xi$	$\left(-{\frac {1}{4}}(1+{\sqrt {5}}),{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(-610,443.19)
$180^{o}=\pi$	$-1$	$0$	$(-1,0)$	(-754,0)
$216^{o}={\frac {6\pi }{5}}$	$-{\frac {\varphi }{2}}$	$-\Xi$	$\left(-{\frac {1}{4}}(1+{\sqrt {5}}),-{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(-610,-443.19)
$252^{o}={\frac {7\pi }{5}}$	$-{\frac {1}{2\varphi }}$	$-\Xi *\varphi$	$\left(-{\frac {1}{4}}({\sqrt {5}}-1),-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(-233,-717.1)
$288^{o}={\frac {8\pi }{5}}$	${\frac {1}{2\varphi }}$	$-\Xi *\varphi$	$\left({\frac {1}{4}}({\sqrt {5}}-1),-{\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}\right)$	(233,-717.1)
$324^{o}={\frac {9\pi }{5}}$	${\frac {\varphi }{2}}$	$-\Xi$	$\left({\frac {1}{4}}(1+{\sqrt {5}}),-{\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}\right)$	(610,-443.19)

Pfad in SVG

<path
   d="m754,0 L610,443.19 233,717.1 -233,717.1
      -610,443.19 -754,0 -610,-443.19
      -233,-717.1 233,-717.1 610,-443.19
      z" />

Sinus/Cosinus von 36° und 72°

	Radius	377	610	233	131	754
Winkelfunktion
$\sin 36^{o}=\sin {\frac {\pi }{5}}$	0,809016994	304,9994069	493,5003666	188,5009597	105,9812263	609,9988138
$\cos 36^{o}=\cos {\frac {\pi }{5}}$	0,587785252	221,5950401	358,5490039	136,9539638	76,99986805	443,1900802
$\sin 72^{o}=\sin {\frac {2\pi }{5}}$	0,309016994	116,4994069	188,5003666	72,00095969	40,48122626	232,9988138
$\cos 72^{o}=\cos {\frac {2\pi }{5}}$	0,951056516	358,5483066	580,1444749	221,5961683	124,5884036	717,0966133

Formeln

aus der deutschsprachigen Wikipedia

Größen eines regelmäßigen Zehnecks mit Seitenlänge (Kantenlänge) a
Inkreisradius	$r_{i}={\frac {a}{2}}{\sqrt {5+2{\sqrt {5}}}}\approx a\cdot 1{,}538842$
Umkreisradius	$r_{u}={\frac {a}{2}}(1+{\sqrt {5}})\approx a\cdot 1{,}618034$
Diagonale über 2 (bzw. 8) Seiten	$d_{2}={\frac {a}{2}}{\sqrt {10+2{\sqrt {5}}}}\approx a\cdot 1{,}902113$
Diagonale über 3 (bzw. 7) Seiten	$d_{3}={\frac {a}{2}}{\sqrt {14+6{\sqrt {5}}}}\approx a\cdot 2{,}618034$
Diagonale über 4 (bzw. 6) Seiten	$d_{4}=a{\sqrt {5+2{\sqrt {5}}}}=2r_{i}\approx a\cdot 3{,}077684$
Diagonale über 5 Seiten	$d=a(1+{\sqrt {5}})=2r_{u}\approx a\cdot 3{,}236068$
Zentriwinkel	$\alpha ={\frac {360^{\circ }}{10}}=36^{\circ }$
Innenwinkel	$\delta =180^{\circ }-\alpha =144^{\circ }$ $\cos \delta =-{\frac {1}{4}}\left(1+{\sqrt {5}}\right)\approx -0{,}8090170$

Annäherung durch Bruchzahlen

Tabelle • Tabellenseite bearbeiten

Zahlenwert	ungefähr	Kettenbruch	sin	cos	Näherung
${\frac {1}{2}}(1+{\sqrt {5}})=\Phi$	$\approx 1.61803398875$	$[1;{\overline {1}}]$	$1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,\ldots$ ${\frac {8}{5}}\diamond {\frac {13}{8}}\diamond {\frac {21}{13}}\diamond {\frac {34}{21}}\diamond \ldots$
${\frac {1}{2}}({\sqrt {5}}-1)=\Phi -1$ $={\frac {1}{\Phi }}$	$\approx 0.61803398875$	$[0;{\overline {1}}]$	${\frac {5}{8}}\diamond {\frac {8}{13}}\diamond {\frac {13}{21}}\diamond {\frac {21}{34}}\diamond \ldots$
${\frac {1}{4}}({\sqrt {5}}-1)={\frac {\Phi -1}{2}}$ $={\frac {1}{2\Phi }}$	$\approx 0.3090169947$	$[0;3,{\overline {4}}]$	$18^{o}={\frac {\pi }{10}}$	$72^{o}={\frac {2\pi }{5}}$	${\frac {4}{13}}\diamond {\frac {17}{55}}\diamond {\frac {72}{233}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5-{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5-{\sqrt {5}})}}$	$\approx 0.58778525229$	$[0;1,1,2,2,1,6,1,56,\ldots ]$	$36^{o}={\frac {\pi }{5}}$	$54^{o}={\frac {3\pi }{10}}$	${\frac {4379}{7450}}\diamond {\frac {77}{131}}$
${\frac {1}{4}}(1+{\sqrt {5}})={\frac {\Phi }{2}}$	$\approx 0.8090169943749474$	$[0;1,{\overline {4}}]$	$54^{o}={\frac {3\pi }{10}}$	$36^{o}={\frac {\pi }{5}}$	${\frac {305}{377}}\diamond {\frac {72}{89}}\diamond {\frac {17}{21}}$
${\frac {\sqrt {2}}{4}}{\sqrt {5+{\sqrt {5}}}}$ $={\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}$ $={\frac {1}{2}}{\sqrt {{\frac {1}{2}}(5+{\sqrt {5}})}}$	$\approx 0.9510565162951536$	$[0;1,19,2,3,6,5,1,1,1,\ldots ]$	$72^{o}={\frac {2\pi }{5}}$	$18^{o}={\frac {\pi }{10}}$	${\frac {855}{899}}\diamond {\frac {136}{143}}\diamond {\frac {39}{41}}$
	$\approx 0.9510565162951536$	$[0;1,19,2,3,6,5,1,1,1,\ldots ]$	$108^{o}={\frac {3\pi }{5}}$	$-162^{o}=-{\frac {9\pi }{10}}$
${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}$	$\approx 0.2588190$	$[0;3,1,6,2,1,30,5,2,9,3,1,\ldots ]$	$15^{o}={\frac {\pi }{12}}$	$75^{o}={\frac {5\pi }{12}}$	${\frac {22}{85}}\diamond {\frac {675}{2608}}$
${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ $={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}$	$\approx 0.96592582$	$[0;1,28,2,1,7,21,1,8,1,\ldots ]$	$75^{o}={\frac {5\pi }{12}}$	$15^{o}={\frac {\pi }{12}}$	${\frac {652}{675}}\diamond {\frac {85}{88}}$
${\frac {\sqrt {2}}{2}}$	$\approx 0.7071067811865475$	$[0;1,{\overline {2}}]$	$45^{o}={\frac {\pi }{4}}$		${\frac {5}{7}}\diamond {\frac {12}{17}}\diamond {\frac {29}{41}}\diamond {\frac {70}{99}}\diamond {\frac {169}{239}}\diamond {\frac {408}{577}}$
${\frac {1}{2}}$			$30^{o}={\frac {\pi }{6}}$	$60^{o}={\frac {\pi }{3}}$
${\frac {\sqrt {3}}{2}}$	$\approx 0.8660254037844386$	$[0;1,{\overline {6,2}}]$	$60^{o}={\frac {\pi }{3}}$	$30^{o}={\frac {\pi }{6}}$	${\frac {84}{97}}\diamond {\frac {181}{209}}\diamond {\frac {1170}{1351}}$
${\sqrt {2}}$	$\approx 1.414213562373095$	$[1;{\overline {2}}]$	${\frac {99}{70}}\diamond {\frac {239}{169}}\diamond {\frac {577}{408}}$
${\sqrt {3}}$	$\approx 1.732050807568877$	$[1;{\overline {1,2}}]$	${\frac {99}{70}}\diamond {\frac {265}{153}}\diamond {\frac {362}{209}}\diamond {\frac {1351}{780}}$
${\sqrt {5}}$	$\approx 2.23606797749979$	$[2;{\overline {4}}]$	${\frac {95}{56}}\diamond {\frac {521}{233}}\diamond {\frac {682}{305}}$
$4{\frac {({\sqrt {2}}-1)}{3}}=\varkappa$	$\approx 0.5522847498307933$	$[0;{\overline {1,1,4,3}}]$	${\frac {5}{9}}\diamond {\frac {16}{29}}\diamond {\frac {21}{38}}\diamond {\frac {37}{67}}\diamond {\frac {169}{306}}\diamond {\frac {544}{985}}$