Decagon

A decagon

A decagon is a shape with 10 sides and 10 corners.

Regular decagon

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

Area

The amount of space a regular decagon takes up is

${\text{Area}}={\frac {5}{2}}a^{2}{\sqrt {5+2{\sqrt {5}}}}.$

a is the length of one of its sides.

An alternative formula is $A=2.5dt$ where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry $d=2t(\cos {54^{\circ }}+\cos {18^{\circ }})$.

Sides

The side of a regular decagon inscribed in a unit circle is ${\tfrac {-1+{\sqrt {5}}}{2}}={\tfrac {1}{\phi }}$, where ϕ is the golden ratio, ${\tfrac {1+{\sqrt {5}}}{2}}$.

Dissection of regular decagon

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. 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 decagon

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 D_5d, [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 pages

• Decagonal number and centered decagonal number, figurate numbers modeled on the decagon
• Decagram, a star polygon with the same vertex positions as the regular decagon

Other websites

• Eric W. Weisstein, Decagon at MathWorld.
• Definition and properties of a decagon With interactive animation