Decagon Facts for Kids

A decagon

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

Regular decagon
- Area
Sides
Dissection of regular decagon
Skew decagon
Related pages
Images for kids
See also

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

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

Eric W. Weisstein, Decagon at MathWorld.

Images for kids

Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center.

Decagon facts for kids

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