File:21-Eck-Näherung.svg

Summary[edit]

Description21-Eck-Näherung.svg	Deutsch: Einundzwanzigeck, Näherungskonstruktion bei gegebenem Umkreisradius English: Icosihenagon, approximation at given circumcircle
Date	13 September 2018
Source	Own work
Author	Petrus3743
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.   This SVG trigonometry uses the path text method.

Näherungsonstruktion[edit]

• Die Methode ähnelt der des Dreizehnecks

Ergebnis[edit]

Bezogen auf den Einheitskreis r = 1 [LE][edit]

• Konstruierte Seitenlänge des 21-Ecks in GeoGebra (Anzeige max. 15 Nachkommastellen) ${\displaystyle a=0{,}298084532352349\;[LE]}$
• Seitenlänge des 21-Ecks ${\displaystyle a_{SOLL}=2\cdot \sin \left({\frac {180^{\circ }}{21}}\right)=0{,}298084532352349\ldots \;[LE]}$
• Absoluter Fehler der konstruierten Seitenlänge

Bis zu den max. angezeigten 15 Nachkommastellen ist der absolute Fehler ${\displaystyle F_{a}=a-a_{SOLL}=0{,}0\;[LE]}$

• Konstruierter Zentriwinkel des 21-Ecks in GeoGebra (Anzeige signifikante 13 Nachkommastellen) ${\displaystyle \mu =17{,}1428571428571^{\circ }}$
• Zentriwinkel des 21-Ecks ${\displaystyle \mu _{SOLL}={\frac {360^{\circ }}{21}}=17{,}{\overline {142857}}^{\circ }}$
• Absoluter Fehler des konstruierten Zentriwinkels

Bis zu den angezeigten signifikanten 13 Nachkommastellen ist der absoluter Fehler ${\displaystyle F_{\mu }=\mu -\mu _{SOLL}=0^{\circ }}$

Beispiel um den Fehler zu verdeutlichen[edit]

Bei einem Umkreisradius r = 1 Mrd. km (das Licht bräuchte für diese Strecke ca. 55 min), wäre der absolute Fehler der konstruierten Seitenlänge < 1 mm.

Approximate construction[edit]

• The method is similar to that of the Tridecagon

Result[edit]

Based on the unit circle r = 1 [unit of length][edit]

• Constructed side length of the icosihenagon in GeoGebra (display max 15 decimal places) ${\displaystyle a=0.298084532352349\;[unit\;of\;length]}$
• Side length of the icosihenagon ${\displaystyle a_{target}=2\cdot \sin \left({\frac {180^{\circ }}{21}}\right)=0.298084532352349\ldots \;[unit\;of\;length]}$
• Absolute error of the constructed side length

Up to the max. displayed 15 decimal places is the absolute error ${\displaystyle F_{a}=a-a_{target}=0.0\;[unit\;of\;length]}$

• Constructed central angle of the icosihenagon in GeoGebra (display significant 13 decimal places) ${\displaystyle \mu =17.1428571428571^{\circ }}$
• Central angle of the icosihenagon ${\displaystyle \mu _{target}={\frac {360^{\circ }}{21}}=17.{\overline {142857}}^{\circ }}$
• Absolute error of the constructed central angle

Up to the indicated significant 13 decimal places is the absolute error ${\displaystyle F_{\mu }=\mu -\mu _{target}=0^{\circ }}$

Example to illustrate the error[edit]

At a circumscribed circle radius r = 1 billion km (the light needed for this distance about 55 minutes), the absolute error of the 1st side would be < 1 mm.

Licensing[edit]

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