 Octagon

For other uses, see Octagon (disambiguation).
Regular octagon
A regular octagonType general type of this shape Edges and vertices 8 Schläfli symbol {8} Coxeter–Dynkin diagrams Symmetry group Dihedral (D_{8}) Area
(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}.
Contents
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
In terms of R (circumradius), the area is
In terms of r (inradius), the area is
These last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be derived as follows:
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 454590 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:
 (approximately)
The area is then as above:
Another simple formula for the area is
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).
Uses of octagons

In many parts of the world, stop signs are in the shape of a regular octagon.
Derived figures

An eightsided 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 8sided 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 contains 6 octagons.

An octagonal antiprism contains two octagons.
Petrie polygons
The octagon is the Petrie polygon for these 12 higher dimensional uniform polytopes, shown in these skew orthogonal projections of in A_{7}, B_{4}, and D_{5} Coxeter planes.
A_{7}
7simplex
Rectified 7simplex
Birectified 7simplex
Trirectified 7simplexB_{4}
16cell
Rectified 16cell
Rectified tesseract
TesseractD_{5}
Trirectified demipenteract
Birectified demipenteract
Rectified demipenteract
DemipenteractSee also
 Octagram
 Octagonal number
 Octagon house
 Oktogon, a major intersection in Budapest, Hungary
 Bumper pool
 Rub el Hizb (also known as Al Quds Star and as Octa Star)
 Smoothed octagon
References
 ^ Shorter Oxford English Dictionary (6th ed.), Oxford University Press, 2007, ISBN 9780199206872
External links
 Octagon Calculator
 Definition and properties of an octagon With interactive animation
 Weisstein, Eric W., "Octagon" from MathWorld.
Regular polygons Listed by number of sides1–10 sides 11–20 sides Others Star polygons Categories: Polygons
 Elementary shapes
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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