Decagon

Regular decagon
Type	Regular polygon
Edges and vertices	10
Schläfli symbol	{10} t{5}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D10)
Internal angle (degrees)	144°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

Gonbad-e Qabus designed as decagon, the tallest pure brick tower in the world.

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 each internal angle equal to 144°. Its Schläfli symbol is {10}.

Regular decagon

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

$A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2.$

An alternative formula is $\scriptstyle A\,=\,2.5dt$ where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry $\scriptstyle d\,=\,2t(\cos{54^\circ}\,+\,\cos{18^\circ})$.

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

A regular decagon is constructible using compass and straightedge:

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

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.

{10/3} Decagram

{10/2} or 2{5}

{10/4} or 2{5/2}

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	Rectified 9-simplex	Trirectified 9-simplex	Quadrirectified 9-simplex	Quintirectified 9-simplex
BC5	5-orthoplex	Rectified 5-orthoplex	Birectified 5-cube	Rectified 5-cube	5-cube
D6	t1(431)	t3(131)	t2(131)	t1(131)	6-demicube (131)
H3	Dodecahedron	Icosahedron	Icosidodecahedron

See also

• decagonal number
• Gambrel

External links

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