# Octagon

﻿
Octagon
Regular octagon

A regular octagon
Type general type of this shape
Edges and vertices 8
Schläfli symbol {8}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D8)
Area $2(1+\sqrt{2})a^2$
$\simeq 4.828427 a^2$ (with a = edge length)
Internal angle (degrees) 135°
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagon (from the Greek okto, eight[1]) is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

## Regular octagon

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon). The area of a regular octagon of side length a is given by

$A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427125\,a^2.$

In terms of R (circumradius), the area is

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

In terms of r (inradius), the area is

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

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

The area of a regular octagon can be computed as a truncated square.

The area can also be derived as follows:

$\,\!A=S^2-a^2,$

where S is the span of the octagon, or the second shortest diagonal; and a 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 length of a side a, the span S is:

$S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a$
$S=2.414a\,$ (approximately)

The area is then as above:

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

Another simple formula for the area is

$\ A=2ad$

where d is the distance between parallel sides (the same as span S in the diagram).

## Construction

A regular octagon is constructible with compass and straightedge.

## Standard coordinates

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

• (±1, ±(1+√2))
• (±(1+√2), ±1).

## Derived figures

### Petrie polygons

The octagon is the Petrie polygon for these 12 higher dimensional uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes.

 A7 B4 D5 7-simplex Rectified 7-simplex Birectified 7-simplex Trirectified 7-simplex 16-cell Rectified 16-cell Rectified tesseract Tesseract Trirectified demipenteract Birectified demipenteract Rectified demipenteract Demipenteract

## References

1. ^ Shorter Oxford English Dictionary (6th ed.), Oxford University Press, 2007, ISBN 978-0-19-920687-2

Wikimedia Foundation. 2010.

Synonyms:

### Look at other dictionaries:

• Octagón — Ring name(s) Dragón Dorado La Amenaza Elegante Octagón Billed height 1.73 m (5 ft 8 in)[1] Billed weight 86 kg (190 lb) …   Wikipedia

• Octagón — Nacimiento Plantilla:Birthday and age Jalapa, Veracruz Nombres artísticos La Amenaza Elegante Octagón …   Wikipedia Español

• Octagon — (englisch „Achteck“) bezeichnet: Octagon (Film), ein US amerikanischer Film von Eric Karson von 1980 mit Chuck Norris und Lee van Cleef Octagon (String Trio of New York), ein Album von String Trio of New York von 1992 Octagon (Bathory ), ein… …   Deutsch Wikipedia

• Octagon — Студийный альбом Bathory Дата выпуска 1 мая 1995 …   Википедия

• Octagon — Álbum de Estudio de Bathory Publicación 17 de octubre de 1995 Grabación 30 de enero al 10 de febrero y del 13 al 23 de marzo de 1995 Género(s) Thrash metal …   Wikipedia Español

• Octagon — Oc ta*gon, n. [Gr. ? eight cornered; okta (for oktw eight) + ? an angle: cf. F. cctogone.] 1. (Geom.) A plane figure of eight sides and eight angles. [1913 Webster] 2. Any structure (as a fortification) or place with eight sides or angles. [1913… …   The Collaborative International Dictionary of English

• octagon — (n.) 1650s, from L. octagonos, from Gk. oktagononos, lit. eight angled, from okta comb. form of okto eight (see EIGHT (Cf. eight)) + gonia angle, related to gony knee (see KNEE (Cf. knee)) …   Etymology dictionary

• octagon — ► NOUN ▪ a plane figure with eight straight sides and eight angles. DERIVATIVES octagonal adjective …   English terms dictionary

• octagon — [äk′tə gän΄] n. [L octagonum < Gr oktagōnos, eight cornered: see OCTA & GON] a plane figure with eight angles and eight sides octagonal [äk′tag′ə nəl] adj. octagonally adv …   English World dictionary

• octagon — UK [ˈɒktəɡən] / US [ˈɑktəˌɡɑn] noun [countable] Word forms octagon : singular octagon plural octagons maths a shape with eight straight sides Derived word: octagonal UK [ɒkˈtæɡən(ə)l] / US [ɑkˈtæɡən(ə)l] adjective …   English dictionary