# Decagon

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

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, 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, $\tfrac{1+\sqrt{5}}{2}$.[4]

### Construction

A regular decagon is constructible using compass and straightedge:[4]

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

A decagram is a 12-sided star polygon, represented by symbol {10/n}. There is one regular star polygon: {10/3}, using the same vertices, but connecting every third points. There are also three compounds: {10/5} is reduced to 5 digons, {10/4} is reduced to 2{5/2} as two pentagrams, {10/2} is reduced to 2{5} as two pentagons.

 t{5}, {10/2} or 2{5}, {10/3}, Decagram {10/4} or 2{5/2}, {10/5} or 5{2}

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated hexagon is a dodecagon, t{5}={10}. A quasitruncated pentagram, inverted as {5/3}, is a decagram, t{5/3}={10/3}.

 t{5}={10} t{5/3}={10/3}

### 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 D6 B5 H3 2H2

9-simplex

(411)

(131)

5-orthoplex

5-cube

{5,3}

{3,5}

r{5,3}

Rhombic triacontahedron

5-5 duopyramid

5-5 duoprism