Decagon

Regular decagon
A regular decagon
Type	Regular polygon
Edges and vertices	10
Schläfli symbol	{10}, t{5}
Coxeter diagram
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.

Regular decagon

A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} 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)

$A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694 t^2.$

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,

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

and it can be written algebraically as

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

Sides

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, $\tfrac{1+\sqrt{5}}{2}$.

Construction

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.

Construction of decagon

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.

Symmetry

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. 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

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.

Regular octagon dissected

With 10 rhombs

With 10 rhombs

Petrie polygons

The regular decagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections in various Coxeter planes: The number of sides in the Petrie polygon is equal to the Coxeter number, h, for each symmetry family.

H3
Dodecahedron	Icosahedron	Icosidodecahedron	Rhombic triacontahedron
A9	D6		B5
9-simplex	411	131	5-orthoplex	5-cube

See also

• Decagonal number and centered decagonal number, figurate numbers modeled on the decagon
• Decagram, a star polygon with the same vertex positions as the regular decagon

External links

• Weisstein, Eric W., "Decagon", MathWorld.
• Definition and properties of a decagon With interactive animation