Decagon

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Regular decagon
Regular polygon 10 annotated.svg
A regular decagon
Type Regular polygon
Edges and vertices 10
Schläfli symbol {10}
t{5}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.png
Symmetry group D10, order 2×10
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.[1] A regular decagon has all sides of equal length and each internal angle equal to 144°.[1] Its Schläfli symbol is {10}.[2]

Regular decagon[edit]

The area of a regular decagon is: (with t = edge length)[3]

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, or the diameter of the decagon's inscribed circle. By simple trigonometry,

d=2t\left(\cos\tfrac{3\pi}{10}+\cos\tfrac{\pi}{10}\right),

and it can be written algebraically as

d=t\sqrt{5+2\sqrt{5}}.

Sides[edit]

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}.[4]

Construction[edit]

A regular decagon is constructible using compass and straightedge:[4]

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[edit]

A decagram is a 12-sided star polygon, represented by symbol {10/n}. There is one regular star polygon: {10/3}, using the same vertices, but connecting every third points. There are also three compounds: {10/5} is reduced to 5 digons, {10/4} is reduced to 2{5/2} as two pentagrams, {10/2} is reduced to 2{5} as two pentagons.

Truncated polygon 10.svg
t{5}, CDel node 1.pngCDel 5.pngCDel node 1.png
Regular star figure 2(5,1).svg
{10/2} or 2{5}, CDel node h3.pngCDel 10.pngCDel node.png
Regular star polygon 10-3.svg
{10/3}, CDel node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node.png
Decagram
Regular star figure 2(5,2).svg
{10/4} or 2{5/2}, CDel node h3.pngCDel 10.pngCDel rat.pngCDel 2x.pngCDel node.png
Regular star figure 5(2,1).svg
{10/5} or 5{2}

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t{5}={10}. A quasitruncated pentagram, inverted as {5/3}, is a decagram, t{5/3}={10/3}.

Vertex-transitive truncations of pentagon
Regular polygon truncation 5 1.svg
t{5}={10}
Regular polygon truncation 5 2.svg Regular polygon truncation 5 3.svg
Regular star truncation 5-3 1.svg
t{5/3}={10/3}
Regular star truncation 5-3 2.svg Regular star truncation 5-3 3.svg

Petrie polygons[edit]

The regular decagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections in various Coxeter planes:

A9 D6 B5 H3 2H2
9-simplex t0.svg
9-simplex
6-cube t5 B5.svg
(411)
6-demicube t0 D6.svg
(131)
5-cube t4.svg
5-orthoplex
5-cube t0.svg
5-cube
Dodecahedron petrie.png
{5,3}
Icosahedron petrie.png
{3,5}
Dodecahedron t1 H3.png
r{5,3}
Dual dodecahedron t1 H3.png
Rhombic triacontahedron
5-5 duopyramid ortho.png
5-5 duopyramid
5-5 duoprism ortho-Dih5.png
5-5 duoprism

See also[edit]

References[edit]

  1. ^ a b Sidebotham, Thomas H. (2003), The A to Z of Mathematics: A Basic Guide, John Wiley & Sons, p. 146, ISBN 9780471461630 .
  2. ^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595 .
  3. ^ The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge, 1850, p. 59 . Note that this source uses a as the edge length and gives the argument of the cotangent as an angle in degrees rather than in radians.
  4. ^ a b Ludlow, Henry H. (1904), Geometric Construction of the Regular Decagon and Pentagon Inscribed in a Circle, The Open Court Publishing Co. .

External links[edit]