Regular decagon
Regular decagon.svg
Edges and vertices 10
Schläfli symbols {10}
t{5}
Coxeter–Dynkin diagrams CDW ring.pngCDW 10.pngCDW dot.png
CDW ring.pngCDW 5.pngCDW ring.png
Symmetry group Dihedral (D10)
Area
(with = edge length)
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 (144°). Its Schläfli symbol is {10}. The area of a regular decagon of side length a is given by

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.svg
9-simplex
9-simplex t1.svg
Rectified 9-simplex
9-simplex t2.svg
Birectified 9-simplex
9-simplex t3.svg
Trirectified 9-simplex
9-simplex t4.svg
Quadrirectified 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

ar:عشاري ast:Decágonu cs:Desetiúhelník eo:Deklatero gl:Decágono it:Decagono ka:დეკაგონი ht:Dekagòn hu:Tízszög mk:Десеттоаголник nl:Tienhoek nn:Dekagon no:Dekagon pl:Dziesięciokąt foremny pt:Decágono sr:Десетоугао sv:Dekagon th:รูปสิบเหลี่ยม

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