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}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel node.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. By simple trigonometry d=2t\left(\cos\tfrac{3\pi}{10}+\cos\tfrac{\pi}{10}\right).

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]

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.

Truncated pentagon.png
A truncated regular pentagon
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[edit]

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
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)
B5 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
2H2 5-5 duopyramid ortho.png
5-5 duopyramid
5-5 duoprism ortho-Dih5.png
5-5 duoprism
H3 Dodecahedron petrie.png
Dodecahedron
Icosahedron petrie.png
Icosahedron
Dodecahedron t1 H3.png
Icosidodecahedron
Dual dodecahedron t1 H3.png
Rhombic triacontahedron

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]