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Decagon

ten-sided shape

Regular decagonEdit

All sides of a regular decagon are the same length. Each corner is 144°. All corners added together equal 1440°.

AreaEdit

The amount of space a regular decagon takes up is

 

a is the length of one of its sides.

An alternative formula is   where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry  .

SidesEdit

The side of a regular decagon inscribed in a unit circle is  , where ϕ is the golden ratio,  .

Dissection of regular decagonEdit

Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. For the regular decagon, m=5, and it can be divided into 10 rhombs, with examples shown below. This decomposition can be seen as 10 of 80 faces in a Petrie polygon projection plane of the 5-cube. A dissection is based on 10 of 30 faces of the rhombic triacontahedron.[1] The list A006245 defines the number of solutions as 62, with 2 orientations for the first symmetric form, and 10 orientations for the other 6.

Regular decagon dissected into 10 rhombi
 
5-cube
             

Skew decagonEdit

3 regular skew zig-zag decagons
{5}#{ } {5/2}#{ } {5/3}#{ }
     
A regular skew decagon is seen as zig-zagging edges of a pentagonal antiprism, a pentagrammic antiprism, and a pentagrammic crossed-antiprism.

A skew decagon is a skew polygon with 10 vertices and edges but not existing on the same plane. The interior of such an decagon is not generally defined. A skew zig-zag decagon has vertices alternating between two parallel planes.

A regular skew decagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew decagon and can be seen in the vertices and side edges of a pentagonal antiprism, pentagrammic antiprism, and pentagrammic crossed-antiprism with the same D5d, [2+,10] symmetry, order 20.

These can also be seen in these 4 convex polyhedra with icosahedral symmetry. The polygons on the perimeter of these projections are regular skew decagons.

Orthogonal projections of polyhedra on 5-fold axes
 
Dodecahedron
 
Icosahedron
 
Icosidodecahedron
 
Rhombic triacontahedron

Related pagesEdit

ReferencesEdit

  1. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

Other websitesEdit