Octagon

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Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8}
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°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

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.

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 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:

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

As any regular sided polygon can be divided into numbers of equal rightangled triangles, the area can also be calculated more simply, in the case of one with an even number of sides, by taking the distance between any two opposite sides (A), dividing by two and then multiplying by the length of one side(B), divided by four and then multiplying by twice the total of the number of sides (N) as follows:

((A/2) * (B/4)) * 2N

Not as, mathematically, pretty as some of the above formulae, but certainly simpler for the layman and also works for any regular polygon by just changing the value of N. (For regular polygons with an uneven number of sides A is calculated as the distance between the point of one angle to the mid point of the side opposite.)

[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
The famous Vichy Pastilles, octagon-shaped candies.

[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.