OCTAGON

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!bgcolor=#e7dcc3 colspan=2|Regular octagon
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|align=center colspan=2|
A regular octagon
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|bgcolor=#e7dcc3|Edges and vertices||8
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|bgcolor=#e7dcc3|Schläfli symbols||{8}
t{4}
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|bgcolor=#e7dcc3|Coxeter–Dynkin diagrams||

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|bgcolor=#e7dcc3|Symmetry group||Dihedral (D₈)
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|bgcolor=#e7dcc3|Area
(with ''t''=edge length)||$2(1+sqrt\{2\})t^2$
$simeq\; 4.828427\; t^2.$
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|bgcolor=#e7dcc3|Internal angle
(degrees)||135°
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In geometry, an 'octagon' is a polygon that has eight sides. Regular octagon is represented by Schläfli symbol {8}.
Century Gothic

Contents

Regular octagons

Uses of octagons

See also

External links

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
:

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

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

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
: $B\; =\; S/(1+sqrt\{2\}).$

Uses of octagons

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In many parts of the world, stop signs are in the shape of a regular octagon.
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An eight-sided star, called an ''octagram'', with Schläfli symbol {8/3} is contained with a regular octagon.
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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.
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The truncated square tiling has 2 octagons around every vertex.
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The truncated cuboctahedron has 6 octagons
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The octagonal prism and octagonal antiprism both contain two octagons.
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See also

★ octagonal number

External links

★ How to find the area of an octagon

★ Definition and properties of an octagon With interactive animation

★