decagon
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dec·a·gon
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n.
A polygon with ten angles and ten sides.
[Medieval Latin decagōnum, from Greek dekagōnon, from neuter of dekagōnos, having ten angles : deka, ten + -gōnos, angled; see -gon.]
decagonal de·cag'o·nal (dĭ-kăg'ə-nəl) adj.decagonally de·cag'o·nal·ly adv.
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Decagon
| Regular decagon | |
|---|---|
 A regular decagon |
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| Type | Regular polygon |
| Edges and vertices | 10 |
| Schläfli symbol | {10} t{5} |
| Coxeter–Dynkin diagram |     |
| Symmetry group | Dihedral (D10) |
| Internal angle (degrees) | 144° |
| Dual polygon | self |
| Properties | convex, cyclic, equilateral, isogonal, isotoxal |
In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°. Its Schläfli symbol is {10}.
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Contents
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Regular decagon
The area of a regular decagon is: (with t = edge length)
An alternative formula is 
where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry 
.
Sides
The side of a regular decagon inscribed in a unit circle is 
, where ϕ is the golden ratio, 
.
Construction
A regular decagon is constructible using compass and straightedge:
An alternative (but similar) method is as follows:
- Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
- Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.
- The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.
Related figures
There is one regular star polygon, the decagram {10/3}, using the same points, but connecting every third points. There are also two compounds: {10/4} is reduced to 2{5/2} as two pentagrams, and {10/2} is reduced to 2{5} as two pentagons.
 {10/3} Decagram |
 {10/2} or 2{5} |
 {10/4} or 2{5/2} |
Petrie polygons
The regular decagon is the Petrie polygon for many higher dimensional polytopes, shown in these skew orthogonal projections in various Coxeter planes:
See also
External links
- Weisstein, Eric W., "Decagon" from MathWorld.
- Definition and properties of a decagon With interactive animation
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Translations:
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Decagon
Dansk (Danish)
n. - tisidet figur
Français (French)
n. - (Géom) décagone
Ελληνική (Greek)
n. - δεκάγωνο
Português (Portuguese)
n. - decágono (m)
Русский (Russian)
десятиугольник
Español (Spanish)
n. - decágono
Svenska (Swedish)
n. - tiohörning
中文（简体）(Chinese (Simplified))
十边形, 十角形
中文（繁體）(Chinese (Traditional))
n. - 十邊形, 十角形
العربيه (Arabic)
(الاسم) شكل بعشرة زوايا وجوانب
עברית (Hebrew)
n. - בעל עשר צלעות, מעושר
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American Heritage Dictionary
The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.
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