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Octagon
From Wikipedia, the free encyclopedia
| Regular octagon | |
|---|---|
 A regular octagon |
|
| Edges and vertices | 8 |
| Schläfli symbols | {8} t{4} |
| Coxeter–Dynkin diagrams |     |
| Symmetry group | Dihedral (D8) |
| Area (with t=edge length) |
  |
| Internal angle (degrees) |
135° |
In geometry, an octagon is a polygon that has eight sides. Regular octagon is represented by Schläfli symbol {8}.
Contents |
A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080°.
The area of a regular octagon of side length a is given by
In terms of R, (circumradius) the area is
In terms of r, (inradius) the area is
Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.
The area may also be found this way:
- A = S2 − B2.
Where S is the span of the octagon, or the second shortest diagonal; and B is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45-45-90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base. Given the span S the length of a side B is
 In many parts of the world, stop signs are in the shape of a regular octagon. |
 Push-button |
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 An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon. |
 The vertex figure of the uniform polyhedron, great dirhombicosidodecahedron is contained within an irregular 8-sided star polygon, with four edges going through its center. |
 The octagonal prism contain two octagons. |
 The truncated square tiling has 2 octagons around every vertex. |
 The truncated cuboctahedron has 6 octagons |
 The octagonal antiprism contain two octagons. |
- How to find the area of an octagon
- Definition and properties of an octagon With interactive animation
- Eric W. Weisstein, Octagon at MathWorld.
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| Triangle · Quadrilateral · Pentagon · Hexagon · Heptagon · Octagon · Enneagon (Nonagon) · Decagon · Hendecagon · Dodecagon · Triskaidecagon · Pentadecagon · Hexadecagon · Heptadecagon · Enneadecagon · Icosagon · Chiliagon · Myriagon |