Decagon
A decagon is a shape with 10 sides and 10 corners.
Regular decagon[change]
All sides of a regular decagon are the same length. Each corner is 144°. All corners added together equal 1440°.
Area[change]
The amount of space a regular decagon takes up is
- <math>\text{Area} = \frac{5}{2} a^2\sqrt{5+2\sqrt{5}}.</math>
a is the length of one of its sides.
An alternative formula is <math>A = 2.5dt</math> where d is the distance between parallel sides, or the height when the decagon stands on one side as base.
By simple trigonometry <math>d = 2t(\cos{54^\circ} + \cos{18^\circ})</math>.
Sides[change]
The side of a regular decagon inscribed in a unit circle is <math>\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}</math>, where ϕ is the golden ratio, <math>\tfrac{1+\sqrt{5}}{2}</math>.
Dissection of regular decagon[change]
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.
| File:5-cube t0.svg 5-cube |
File:Sun decagon.svg | File:Sun2 decagon.svg | File:Dart2 decagon.svg | File:Halfsun decagon.svg | File:Dart decagon.svg | File:Dart decagon ccw.svg | File:Cartwheel decagon.svg |
Skew decagon[change]
| {5}#{ } | {5/2}#{ } | {5/3}#{ } |
|---|---|---|
| File:Regular skew polygon in pentagonal antiprism.png | File:Regular skew polygon in pentagrammic antiprism.png | File:Regular skew polygon in pentagrammic crossed-antiprism.png |
| 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.
| File:Dodecahedron petrie.png Dodecahedron |
File:Icosahedron petrie.svg Icosahedron |
File:Dodecahedron t1 H3.png Icosidodecahedron |
File:Dual dodecahedron t1 H3.png Rhombic triacontahedron |
Related pages[change]
- 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
References[change]
Other websites[change]
- Eric W. Weisstein, Decagon at MathWorld.
- Definition and properties of a decagon With interactive animation
