Decagon

Regular decagon

A regular decagon
Type Regular polygon
Edges and vertices 10
Schläfli symbol {10}
Coxeter diagram
Symmetry group D10, order 2×10
Internal angle (degrees) 144°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal
Gonbad-e Qabus, the tallest pure brick tower in the world, is built on a decagonal plan.

In geometry, a decagon is any polygon with ten sides and ten angles. A regular decagon has 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)

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

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.

 A truncated regular pentagon {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: