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Regular octagon | |||
---|---|---|---|

A regular octagon | |||

Edges and vertices | 8 | ||

Schläfli symbols | {8} t{4} | ||

Coxeter–Dynkin diagrams | >- | Symmetry group | Dihedral (D_{8}) |

Area (with t=edge length) | $2(1+sqrt\{2\})t^2$ $simeq\; 4.828427\; t^2.$ | ||

Internal angle (degrees) | 135° |

In geometry, an octagon is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

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

- $A\; =\; 2\; cot\; frac\{pi\}\{8\}\; a^2\; =\; 2(1+sqrt\{2\})a^2\; simeq\; 4.828427,a^2.$

In terms of $R$, (circumradius) the area is

- $A\; =\; 4\; sin\; frac\{pi\}\{4\}\; R^2\; =\; 2sqrt\{2\}R^2\; simeq\; 2.828427,R^2.$

In terms of $r$, (inradius) the area is

- $A\; =\; 8\; tan\; frac\{pi\}\{8\}\; r^2\; =\; 8(sqrt\{2\}-1)r^2\; simeq\; 3.3137085,r^2.$

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.

The area may also be found this way:

- $,!A=S^2-B^2.$

Given the span $S$, the length of a side $B$ is

- $S=frac\{B\}\{sqrt\{2\}\}+B+frac\{B\}\{sqrt\{2\}\}=(1+sqrt\{2\})B$

- $S=2.414B$

The area, then, is

- $A=((1+sqrt\{2\})B)^2-B^2=2(1+sqrt\{2\})B^2.$

In many parts of the world, stop signs are in the shape of a regular octagon. | Push-button | |

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. | An octagonal prism contains two octagons. |

The truncated square tiling has 2 octagons around every vertex. | The truncated cuboctahedron has 6 octagons | An octagonal antiprism contains two octagons. |

- How to find the area of an octagon
- Definition and properties of an octagon With interactive animation

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Last updated on Saturday October 11, 2008 at 08:04:56 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 08:04:56 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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