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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 Dihedral (D10), order 2×10
Internal angle (degrees) 144°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a decagon is a 10-sided polygon or 10-gon.[1]

Regular decagon[edit]

A regular decagon has all sides of equal length and each internal angle will always be equal to 144°.[1] Its Schläfli symbol is {10} [2] and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.

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 7.694 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,


and it can be written algebraically as



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]


As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon.[4]

Regular Decagon Inscribed in a Circle.gif
Construction of decagon
Regular Pentagon Inscribed in a Circle 240px.gif
Construction of pentagon

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.


Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.

The regular decagon has Dih10 symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih5, Dih2, and Dih1, and 4 cyclic group symmetries: Z10, Z5, Z2, and Z1.

These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[5] Full symmetry of the regular form is r20 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g10 subgroup has no degrees of freedom but can seen as directed edges.

The highest symmetry irregular decagons are d10, a isogonal decagon constructed by five mirrors which can alternate long and short edges, and p10, an isotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular decagon.

Dissection of regular decagon[edit]

Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. For the decagon, m=5, and it can be divided into 10 rhombs, with one example shown below. This decomposition can be seen as 10 of 80 faces in a Petrie polygon projection plane of the 5-cube. A second dissection is based on 10 of 30 faces of the rhombic triacontahedron.[6]

Regular octagon dissected
Rhombic dissected decagon.png
With 10 rhombs
Rhomb dissected dodecagon2.png
With 10 rhombs

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:[7] The number of sides in the Petrie polygon is equal to the Coxeter number, h, for each symmetry family.

Dodecahedron petrie.png
Icosahedron petrie.png
Dodecahedron t1 H3.png
Dual dodecahedron t1 H3.png
Rhombic triacontahedron
A9 D6 B5
9-simplex t0.svg
6-cube t5 B5.svg
6-demicube t0 D6.svg
5-cube t4.svg
5-cube t0.svg

See also[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. .
  5. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
  6. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  7. ^ Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223-226.

External links[edit]