# Decagon

1,013pages on
this wiki
Regular decagon
Edges and vertices10
Schläfli symbols{10}
t{5}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (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: