Dodekagono
{{Regular polygon db|Regular polygon stat table|p12}}
Sa [[heometriya]], an '''dodekagono''' ([[Ingles]]: ''dodecagon'') o 12-gon iyo an maski anong [[poligono]] na igwang kagduwa (dose) na gilid.

== Regular na dodekagono ==

An regular na dodekagono iyo sarong pigura na igwang gilid na may pararehas na laba asin mga panlaog na angulo na may pararehas na sukol. Igwa ining mga kagduwang (12) linya nin replektibong simetriya asin rotasyonal na simetriya nin order na 12. An regular na dodekagono iyo pigrerepresenta nin simbolong Schläfli na {12} asin pwede man mahaman bilang sarong trunkadong [[heksagono]], t{6}, o sarong ''twice-truncated'' na [[triangulo]], tt{3}. An panlaog na angulo sa kada bertiko kan regular na dodekagono iyo 150°.

=== Hiwas ===
An hiwas kan sarong regular na dodekagono na may gilid sa laba na ''a'' iyo makukua sa paagi kan: 

: <math>\begin{align} A & = 3 \cot\left(\frac{\pi}{12} \right) a^2 = 
                     3 \left(2+\sqrt{3} \right) a^2 \\
                 & \simeq 11.19615242\,a^2
 \end{align}</math>

Asin sa termino nin [[apothem]] na ''r'', an hiwas iyo:

: <math>\begin{align} A & = 12 \tan\left(\frac{\pi}{12}\right) r^2 = 
                     12 \left(2-\sqrt{3} \right) r^2 \\
                 & \simeq 3.2153903\,r^2
 \end{align}</math>

Sa termino kan sirkumradiyus (''circumradius'') na ''R'', an hiwas iyo:<ref>See also Kürschák's geometric proof on [http://demonstrations.wolfram.com/KurschaksDodecagon/ the Wolfram Demonstration Project]</ref>

: <math>A = 6 \sin\left(\frac{\pi}{6}\right) R^2 = 3 R^2</math>

An span na ''S'' kan dodekagono iyo an distansya sa tahaw kan duwang paralelong gilid asin iyo parejas sa duwang apothem. Sarong simpleng pormula para sa hiwas (gamit an laba kan gili asin an span) iyo: 

: <math>A = 3aS</math>

Mabeberipika ini gamit an trigonometrikong relasyon na:

: <math>S = a(1+ 2\cos{30^{\circ}} + 2\cos{60^{\circ}})</math>

=== Perimetro ===
An perimetro kan regular na dodekagono gamit an sirkumradiyus iyo:<ref>''Plane Geometry: Experiment, Classification, Discovery, Application'' by Clarence Addison Willis B., (1922) Blakiston's Son & Company, p. 249 [https://books.google.com/books?id=vDIAAAAAYAAJ&pg=PA249&dq=6.211657+polygon+of+12+sides&hl=en&sa=X&ei=p7niU_yRLsrIsATy3ICIAg&ved=0CCkQ6AEwAA#v=onepage&q=6.211657%20polygon%20of%2012%20sides&f=false]</ref> 

: <math>\begin{align} p & = 24R \tan\left(\frac{\pi}{12}\right) = 12R \sqrt{2 - \sqrt{3}}\\
& \simeq 6.21165708246\,R
\end{align}</math>

An perimtero gamit an apothem iyo:

: <math>\begin{align} p & = 24r \tan\left(\frac{\pi}{12}\right) = 24r(2-\sqrt{3})\\
& \simeq 6.43078061835\,r
\end{align}</math>

An coefficient na ini iyo doble kan coefficient na mahihiling sa ekwasyong apothem para sa hiwas.<ref>''Elements of geometry''
 by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 [https://books.google.com/books?id=qYBTAAAAYAAJ&pg=PA243&dq=6.43078&hl=en&sa=X&ei=tlThU9qKMtTboASlsoDgDw&ved=0CCoQ6AEwAg#v=onepage&q=6.43078&f=false]</ref>

==Panluwas na takod==
{{commons category|12-gons}}
*{{MathWorld |urlname=Dodecagon |title=Dodecagon}}
*[http://www.cut-the-knot.org/Curriculum/Geometry/KurschakTile.shtml Kürschak's Tile and Theorem]
*[http://www.mathopenref.com/dodecagon.html Definition and properties of a dodecagon] Igwa nin inter-aktibong animasyon
*[https://www.mathedpage.org/manipulatives/pattern-blocks/dodecagons The regular dodecagon in the classroom], using [https://www.mathedpage.org/manipulatives/pattern-blocks/index.html pattern blocks]

== Toltolan ==
{{reflist}}
{{matematika-poon}}
[[Kategorya:Mga hugis]]
Sa heometriya, an dodekagono (Ingles: dodecagon) o 12-gon iyo an maski anong poligono na igwang kagduwa (dose) na gilid.

 Regular na dodekagono 

An regular na dodekagono iyo sarong pigura na igwang gilid na may pararehas na laba asin mga panlaog na angulo na may pararehas na sukol. Igwa ining mga kagduwang (12) linya nin replektibong simetriya asin rotasyonal na simetriya nin order na 12. An regular na dodekagono iyo pigrerepresenta nin simbolong Schläfli na {12} asin pwede man mahaman bilang sarong trunkadong heksagono, t{6}, o sarong twice-truncated na triangulo, tt{3}. An panlaog na angulo sa kada bertiko kan regular na dodekagono iyo 150°.

 Hiwas 
An hiwas kan sarong regular na dodekagono na may gilid sa laba na a iyo makukua sa paagi kan: 

 

Asin sa termino nin apothem na r, an hiwas iyo:

 

Sa termino kan sirkumradiyus (circumradius) na R, an hiwas iyo:See also Kürschák's geometric proof on the Wolfram Demonstration Project

 

An span na S kan dodekagono iyo an distansya sa tahaw kan duwang paralelong gilid asin iyo parejas sa duwang apothem. Sarong simpleng pormula para sa hiwas (gamit an laba kan gili asin an span) iyo: 

 

Mabeberipika ini gamit an trigonometrikong relasyon na:

 

 Perimetro 
An perimetro kan regular na dodekagono gamit an sirkumradiyus iyo:Plane Geometry: Experiment, Classification, Discovery, Application by Clarence Addison Willis B., (1922) Blakiston's Son & Company, p. 249  

 

An perimtero gamit an apothem iyo:

 

An coefficient na ini iyo doble kan coefficient na mahihiling sa ekwasyong apothem para sa hiwas.Elements of geometry
 by John Playfair, William Wallace, John Davidsons, (1814) Bell & Bradfute, p. 243 

Panluwas na takod

Kürschak's Tile and Theorem
Definition and properties of a dodecagon Igwa nin inter-aktibong animasyon
The regular dodecagon in the classroom, using pattern blocks

 Toltolan 

Kategorya:Mga hugis