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| {{Odd polygon db|Odd polygon stat table|p11}}
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| In [[geometry]], a '''hendecagon''' (also '''undecagon'''<ref>[http://www.mathopenref.com/undecagon.html Undecagon Definition – Math Open Reference]</ref>) is an 11-sided [[polygon]]. (The name ''hendecagon'', from Greek ''hendeka'' "eleven" and ''gon–'' "corner", is often preferred to the hybrid ''undecagon'', whose first syllable ''un–'' is Latin for "one".<ref>[http://mathworld.wolfram.com/Hendecagon.html Hendecagon – from Wolfram MathWorld<!--Bot generated title-->]</ref>)
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| ==Regular hendecagon==
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| A regular hendecagon has [[internal angle]]s of 147.<span style="text-decoration: overline">27</span> [[degree (angle)|degrees]].<ref>{{citation|title=Glencoe mathematics: applications and connections|first=Kay|last=McClain|publisher=Glencoe/McGraw-Hill|year=1998|isbn=9780028330549|page=357}}.</ref> The area of a regular hendecagon with side length ''a'' is given by<ref>{{citation|title=Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation|first=Elias|last=Loomis|publisher=Harper|year=1886|page=72|url=http://books.google.com/books?id=eqELAAAAYAAJ&pg=PA72}}.</ref>
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| :<math>A = \frac{11}{4}a^2 \cot \frac{\pi}{11} \simeq 9.36564\,a^2.</math>
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| A regular hendecagon is not [[Constructible polygon|constructible]] with [[compass and straightedge]].<ref>As [[Carl Friedrich Gauss|Gauss]] proved, a polygon with a prime number ''p'' of sides can be constructed if and only if ''p'' − 1 is a [[power of two]], not true for 11. See {{citation|title=Mathematical Thought From Ancient to Modern Times|volume=2|first=Morris|last=Kline|authorlink=Morris Kline|publisher=Oxford University Press|year=1990|isbn=9780199840427|pages=753–754|url=http://books.google.com/books?id=VOcUBvbUXlEC&pg=PA753}}.</ref> Because 11 is not a [[Pierpont prime]], construction of a regular hendecagon is still impossible [[neusis construction|even with the usage of an angle trisector]].
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| Close approximations to the regular hendecagon can be constructed, however. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a [[unit circle]] as being 14/25 units long.<ref>{{citation|title=A History of Greek Mathematics: From Aristarchus to Diophantus|first=Sir Thomas Little|last=Heath|authorlink=Thomas Little Heath|publisher=The Clarendon Press|year=1921|page=329|url=http://books.google.com/books?id=7DDQAAAAMAAJ&pg=PA329}}.</ref> The animation below shows another approximation.
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| [[File:Approximated_Hendecagon_Inscribed_in_a_Circle.gif|center|Costruzione approssimata dell'endecagono regolare]]
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| ==Use in coinage==
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| The [[Canadian dollar]] coin, the [[loonie]], is similar to, but not exactly, a regular [[hendecagonal prism]],<ref>{{citation|title=A $1 problem|first=Michael J.|last=Mossinghoff|journal=[[American Mathematical Monthly]]|volume=113|issue=5|year=2006|pages=385–402|jstor=27641947|url=http://www.maa.org/sites/default/files/images/upload_library/22/Ford/mossinghoff385.pdf}}</ref> as are the Indian 2-[[Indian rupee|rupee]] coin<ref>{{citation|title=2013 Standard Catalog of World Coins 2001 to Date|first1=George S.|last1=Cuhaj|first2=Thomas|last2=Michael|publisher=Krause Publications|year=2012|isbn=9781440229657|page=402|url=http://books.google.com/books?id=jI5QOJGScHgC&pg=PA402}}.</ref> and several other lesser-used coins of other nations.<ref>{{citation|title=Unusual World Coins|first1=George S.|last1=Cuhaj|first2=Thomas|last2=Michael|edition=6th|publisher=Krause Publications|year=2011|isbn=9781440217128|pages=23, 222, 233, 526}}.</ref> The cross-section of a loonie is actually a [[Reuleaux hendecagon]].
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| ==Related shapes==
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| The hendecagon shares the same set of 11 vertices with four regular [[hendecagram]]s, {11/2}, {11/3}, {11/4}, {11/5}.
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| The regular hendecagon is the [[Petrie polygon]] for 10-dimensional uniform polytopes of the simplex family, projected in a skew [[orthogonal projection]].<ref>[[Coxeter]], H. S. M. ''Petrie Polygons.'' [[Regular Polytopes (book)|Regular Polytopes]], 3rd ed. New York: Dover, 1973. (sec 2.6 ''Petrie Polygons'' pp. 24–25)</ref><ref>{{citation |first=James E. |last=Humphreys |title=Reflection Groups and Coxeter Groups |pages=80 (Section 3.16, ''Coxeter Elements'', table 2, Coxeter number for A<sub>n</sub> is ''n+1'') |publisher=[[Cambridge University Press]] |year=1992 |isbn=978-0-521-43613-7 |url=http://books.google.com/?id=ODfjmOeNLMUC }}</ref>
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| {| class=wikitable
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| |- align=center
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| |[[File:10-simplex t0.svg|150px]]<br>[[10-simplex]]
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| |[[File:10-simplex t1.svg|150px]]<br>[[Rectified 10-simplex]]
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| |[[File:10-simplex t2.svg|150px]]<br>[[Birectified 10-simplex]]
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| |[[File:10-simplex t3.svg|150px]]<br>[[Trirectified 10-simplex]]
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| |[[File:10-simplex t4.svg|150px]]<br>[[Quadrirectified 10-simplex]]
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| |}
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://www.mathopenref.com/undecagon.html Properties of an Undecagon (hendecagon)] With interactive animation
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| *{{MathWorld |title=Hendecagon |urlname=Hendecagon}}
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| {{Polygons}}
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| [[Category:Polygons]]
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