# Decagon

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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 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. See File:Regular Decagon Inscribed in a Circle.gif. [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.

## Petrie polygons

The regular decagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections in various Coxeter planes:[5] 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

## References

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. ^ Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223-226.