Octagon

Regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8}; t{4}
Coxeter–Dynkin diagram	;
Symmetry group	Dihedral (D8)
Area	; (with a = edge length)
Internal angle (degrees)	135°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

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For other uses, see Octagon (disambiguation).

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}.

[edit] 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 (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:

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

$S \approx 2.414a\,$

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).

[edit] Construction

A regular octagon is constructible with compass and straightedge.

[edit] 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).

[edit] Uses of octagons

In many parts of the world, stop signs are in the shape of a regular octagon.
Umbrellas often have an octagonal outline.
The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.
The street & block layout of Barcelona's Eixample district is based on non-regular octagons
Vichy Pastilles, octagon-shaped candies.
Janggi uses octagonal pieces.

[edit] Derived figures

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 contains 6 octagons.
An octagonal antiprism contains two octagons.

[edit] 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	7-simplex	Rectified 7-simplex	Birectified 7-simplex	Trirectified 7-simplex
B₄	16-cell	Rectified 16-cell	Rectified tesseract	Tesseract
D₅	Trirectified demipenteract	Birectified demipenteract	Rectified demipenteract	Demipenteract