Definitions

# Octagon

[ok-tuh-gon, -guhn]
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)
$2\left(1+sqrt\left\{2\right\}\right)t^2$
$simeq 4.828427 t^2.$
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}.

## Regular octagons

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

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

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

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

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

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

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

The area may also be found this way:

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

$S=frac\left\{B\right\}\left\{sqrt\left\{2\right\}\right\}+B+frac\left\{B\right\}\left\{sqrt\left\{2\right\}\right\}=\left(1+sqrt\left\{2\right\}\right)B$
$S=2.414B$

The area, then, is

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

## Uses of octagons

 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.

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