Decagon

Regular decagon
Edges and vertices	10
Schläfli symbols	{10} t{5}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D10)
Area (with $a$ = edge length)	$\begin{align}A&=\frac52\cot\left(\frac{\pi}{10}\right)a^2\\ &=\frac{5\sqrt{5+2\sqrt5}}{2}a^2\\ &\approx7.69a^2\end{align}$
Internal angle (degrees)	144°

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 all internal angles equal to $\frac{4\pi}{5}$ (144°). Its Schläfli symbol is {10}. The area of a regular decagon of side length a is given by

$A=\frac52\cot\left(\frac{\pi}{10}\right)a^2=\frac{5\sqrt{5+2\sqrt5}}{2}a^2\approx7.69a^2$

Construction

A regular decagon is constructible with a compass and straightedge.

1. Complete steps 1 through 6 of constructing a pentagon.
2. Extend a line from each corner of the pentagon through the center of the circle made in step 1 of constructing a pentagon to the opposite side of that same circle.
3. The five corners of the pentagon constitute every other corner of the decagon. The remaining five corners of the decagon are those points where the lines of step 2 cross the original circle (but not a pentagon corner).

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	Birectified 9-simplex	Trirectified 9-simplex	Quadrirectified 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

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