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Decagon

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Regular decagon
Regular decagon
Edges and vertices10
Schläfli symbols{10}
t{5}
Coxeter–Dynkin diagramsCDW ringCDW 10CDW dot
CDW ringCDW 5CDW ring
Symmetry groupDihedral (D10)
Area
(with a = edge length)
\begin{align}A&=\frac52\cot\left(\frac{\pi}{10}\right)a^2\\
&=\frac{5\sqrt{5+2\sqrt5}}{2}a^2\\
&\approx7.69a^2\end{align}
Internal angle
(degrees)
144°

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 all internal angles equal to \frac{4\pi}{5} (144°). Its Schläfli symbol is {10}. The area of a regular decagon of side length a is given by

A=\frac52\cot\left(\frac{\pi}{10}\right)a^2=\frac{5\sqrt{5+2\sqrt5}}{2}a^2\approx7.69a^2

Construction

A regular decagon is constructible with a compass and straightedge.

  1. Complete steps 1 through 6 of constructing a pentagon.
  2. Extend a line from each corner of the pentagon through the center of the circle made in step 1 of constructing a pentagon to the opposite side of that same circle.
  3. The five corners of the pentagon constitute every other corner of the decagon. The remaining five corners of the decagon are those points where the lines of step 2 cross the original circle (but not a pentagon corner).


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


See also

External links

ar:عشاري

ast:Decágonu cs:Desetiúhelníkeo:Deklaterogl:Decágono it:Decagono ka:დეკაგონი ht:Dekagòn hu:Tízszög mk:Десеттоаголникnl:Tienhoek nn:Dekagon no:Dekagon pl:Dziesięciokąt foremny pt:Decágonosr:Десетоугаоsv:Dekagon th:รูปสิบเหลี่ยม

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