Dodecagon

The area of a regular dodecagon of side length a is given by:

${\displaystyle {\begin{aligned}A&=3\cot \left({\frac {\pi }{12}}\right)a^{2}=3\left(2+{\sqrt {3}}\right)a^{2}\\&\simeq 11.19615242\,a^{2}\end{aligned}}}$

And in terms of the apothem r (see also inscribed figure), the area is:

${\displaystyle {\begin{aligned}A&=12\tan \left({\frac {\pi }{12}}\right)r^{2}=12\left(2-{\sqrt {3}}\right)r^{2}\\&\simeq 3.2153903\,r^{2}\end{aligned}}}$

In terms of the circumradius R, the area is:[1]

${\displaystyle A=6\sin \left({\frac {\pi }{6}}\right)R^{2}=3R^{2}}$

The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) is:

${\displaystyle A=3aS}$

This can be verified with the trigonometric relationship:

${\displaystyle S=a(1+2\cos {30^{\circ }}+2\cos {60^{\circ }})}$

The perimeter of a regular dodecagon in terms of circumradius is:[2]

${\displaystyle {\begin{aligned}p&=24R\tan \left({\frac {\pi }{12}}\right)=12R{\sqrt {2-{\sqrt {3}}}}\\&\simeq 6.21165708246\,R\end{aligned}}}$

The perimeter in terms of apothem is:

${\displaystyle {\begin{aligned}p&=24r\tan \left({\frac {\pi }{12}}\right)=24r(2-{\sqrt {3}})\\&\simeq 6.43078061835\,r\end{aligned}}}$

This coefficient is double the coefficient found in the apothem equation for area.[3]

A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways:

3.12.12	4.6.12	3.3.4.12	3.4.3.12

Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration:

1-uniform		2-uniform
3.12.12	4.6.12	3.12.12; 3.4.3.12

The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. In 6 dimensions 6-cube, 6-orthoplex, 2₂₁, 1₂₂. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.

Regular skew dodecagons in higher dimensions
E6		F4		2G2 (4D)
221	122	24-cell	Snub 24-cell	6-6 duopyramid	{6}×{6}
A11	D7		B6		4A2
11-simplex	(411)	141	6-orthoplex	6-cube	{3}×{3}×{3}×{3}