Dodecagon

The constructible regular 12-sided Polygon with Schläfli Symbol $\{12\}$. The Inradius $r$, Circumradius $R$, and Area $A$ can be computed directly from the formulas for a general regular Polygon with side length $s$ and $n=12$ sides,

$$r = {\textstyle{1\over 2}}s\cot\left({\pi\over 12}\right)={\textstyle{1\over 2}}(2+\sqrt{3})s$$ (1)

$$R = {\textstyle{1\over 2}}s\cot\left({\pi\over 12}\right)={\textstyle{1\over 2}}(\sqrt{2}+\sqrt{6}\,)s$$ (2)

$$A = {\textstyle{1\over 4}}n s^2\cot\left({\pi\over 12}\right)=3(2+\sqrt{3}\,)s^2.$$ (3)

A Plane Perpendicular to a $C_5$ axis of a Dodecahedron or Icosahedron cuts the solid in a regular Decagonal Cross-Section (Holden 1991, pp. 24-25).

The Greek, Latin, and Maltese Crosses are all irregular dodecagons.

See also Decagon, Dodecagram, Dodecahedron, Greek Cross, Latin Cross, Maltese Cross, Trigonometry Values Pi/12, Undecagon

References

Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.