Octagon

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Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8} t{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D₈)
Area (with a=edge length)	$2(1+\sqrt{2})a^2$ $\simeq 4.828427 a^2.$
Internal angle (degrees)	135°

For other uses, see Octagon (disambiguation).

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

[edit] Regular octagons

A regular octagon is constructible with compass and straightedge. To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 through 10.

A regular octagon is always 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

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

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

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

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

An octagon inset in a square.

The area can also be derived as folllows:

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

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 span $S$ , the length of a side $a$ is:

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

$S=2.414a\,$

The area, is then as above:

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

[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.

[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.