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Concept
Dodecagon
Summary
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol {12} and can be constructed as a truncated hexagon, t{6}, or a twice-truncated triangle, tt{3}. The internal angle at each vertex of a regular dodecagon is 150°.
The area of a regular dodecagon of side length a is given by:
And in terms of the apothem r (see also inscribed figure), the area is:
In terms of the circumradius R, the area is:
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:
This can be verified with the trigonometric relationship:
The perimeter of a regular dodecagon in terms of circumradius is:
The perimeter in terms of apothem is:
This coefficient is double the coefficient found in the apothem equation for area.
As 12 = 22 × 3, regular dodecagon is constructible using compass-and-straightedge construction:
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular dodecagon, m=6, and it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs. This decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. The sequence OEIS sequence defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection.
One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons. They are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles.
Official source
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