Hendecagon (11-sided polygon)

Geometric properties

Key measurements for a regular hendecagon follow from its elevenfold symmetry. The central angle between adjacent vertices is 360/11 ≈ 32.7273°, and each interior angle measures 180(11-2)/11 = 147.2727...°. The symmetry group is the dihedral group D11, of order 22, combining rotations and reflections that map the figure onto itself.

Constructibility and algebraic background

The regular hendecagon cannot be constructed with classical straightedge and compass: 11 is not a Fermat prime and so the elevenfold division of the circle is not obtainable by the simple constructions that work for triangles, pentagons and some other n-gons. Its vertices correspond to the 11th roots of unity in the complex plane, and the minimal polynomial involved has degree 10, which links the hendecagon to cyclotomic and field theory in algebra.

Star figures and variations

Besides the convex regular 11-gon, there are several regular star polygons — hendecagrams — formed by connecting every k-th vertex for k = 2,3,4,5. For a prime 11 there are five distinct regular hendecagrams often written as {11/2}, {11/3}, {11/4}, and {11/5}, together with their mirror images. Irregular hendecagons can take many shapes, convex or self-intersecting.

History, uses and examples

Although not as prominent historically as polygons with small numbers of sides, the hendecagon occurs in decorative motifs, tiling experiments, and modern design where an elevenfold aesthetic is desired. In mathematics it serves as a standard example when illustrating limits of compass-and-straightedge methods, properties of cyclotomic polynomials, and dihedral symmetry. Approximate geometric constructions and numerical methods can produce accurate representations for practical applications.

For an introductory visual or technical reference, see related material. Notable distinctions include the contrast between regular convex hendecagons and the family of star hendecagrams, and the algebraic consequence that exact Euclidean construction is impossible with classical tools.