Decagon

Decagon
Regular decagon
Regular polygon 10.svg
Edges and vertices 10
Schläfli symbol {10}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 10.pngCDel node.png
Symmetry group Dihedral (D10)
Internal angle (degrees) 144°
Properties convex, cyclic, equilateral, isogonal, isotoxal
Gonbad-e Qabus designed as decagon, the tallest pure brick tower in the world.

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}.

Contents

Regular decagon

The area of a regular decagon is: (with t = edge length)

A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}}  \simeq 7.694208843 t^2.

An alternative formula is \ A = 2.5dt where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry d = 2t(cos54° + cos18°).

Sides

The side of a regular decagon inscribed in a unit circle is \tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}, where ϕ is the golden ratio, \tfrac{1+\sqrt{5}}{2}.

Construction

A regular decagon is constructible using compass and straightedge:

Construction of a regular decagon

An alternative (but similar) method is as follows:

  1. Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
  2. 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.
  3. 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.

Decagram 10 3.png
{10/3}
Decagram
Decagram 10 2.png
{10/2} or 2{5}
Decagram 10 4.png
{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:

A9 9-simplex t0.svg
9-simplex
9-simplex t1.svg
Rectified 9-simplex
9-simplex t2.svg
Trirectified 9-simplex
9-simplex t3.svg
Quadrirectified 9-simplex
9-simplex t4.svg
Quintirectified 9-simplex
BC5 5-cube t4.svg
5-orthoplex
5-cube t3.svg
Rectified 5-orthoplex
5-cube t2.svg
Birectified 5-cube
5-cube t1.svg
Rectified 5-cube
5-cube t0.svg
5-cube
D6 6-cube t5 B5.svg
t1(431)
6-cube t4 B5.svg
t3(131)
6-cube t3 B5.svg
t2(131)
6-demicube t1 D6.svg
t1(131)
6-demicube t0 D6.svg
6-demicube
(131)
H3 Dodecahedron petrie.png
Dodecahedron
Icosahedron petrie.png
Icosahedron
Dodecahedron t1 H3.png
Icosidodecahedron

See also

External links


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Look at other dictionaries:

  • decagon — DECAGÓN, decagoane, s.n. Poligon cu zece laturi. – Din fr. décagone. Trimis de claudia, 13.09.2007. Sursa: DEX 98  decagón s. n., pl. decagoáne Trimis de siveco, 10.08.2004. Sursa: Dicţionar ortografic  DECAG//ÓN decagonoáne n. Poligon cu zece… …   Dicționar Român

  • Decagon — Dec a*gon, n. [Pref. deca + Gr. ? a corner or angle: cf. F. d[ e]cagone.] (Geom.) A plane figure having ten sides and ten angles; any figure having ten angles. A regular decagon is one that has all its sides and angles equal. [1913 Webster] …   The Collaborative International Dictionary of English

  • decagon — early 17c., from Mod.L. decagonum, from Gk. dekagonon, from Gk. deka ten (see TEN (Cf. ten)) + gonia corner, angle (see KNEE (Cf. knee)) …   Etymology dictionary

  • decagon — ► NOUN ▪ a plane figure with ten straight sides and angles …   English terms dictionary

  • decagon — [dek′ə gän΄] n. [ML decagonum: see DECA & GON] a polygon with ten sides and ten angles decagonal [dek ag′ə nəl] adj …   English World dictionary

  • decagon —    A closed two dimensional polygon bounded by ten straight line segments. The formula with which to find an equilateral decagon s area is 7.6942 times the length of one side squared. Also see mathematics, radial, and shape …   Glossary of Art Terms

  • Decagon Devices — is an American corporation that designs and markets scientific instruments, sensors, data loggers and accessories. Decagon was founded in 1988 by Dr. Gaylon Campbell. Currently, Decagon designs and manufactures several different types of… …   Wikipedia

  • decagon — noun Etymology: New Latin decagonum, from Greek dekagōnon, from deka deca + gōnon gon Date: circa 1639 a plane polygon of 10 angles and 10 sides …   New Collegiate Dictionary

  • decagon — decagonal /deuh kag euh nl/, adj. /dek euh gon /, n. Geom. a polygon having ten angles and ten sides. [1565 75; < ML decagonum. See DECA , GON] * * * …   Universalium

  • decagon — noun A polygon with ten sides and ten angles. See Also: decagonal …   Wiktionary

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