Octagon

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Regular octagon
Image:Octagon.svg A regular octagon
Edges and vertices	8
Schläfli symbols	{8} t{4}
Coxeter–Dynkin diagrams	Image:CDW ring.pngImage:CDW 8.pngImage:CDW dot.png Image:CDW ring.pngImage:CDW 4.pngImage:CDW ring.png
Symmetry group	Dihedral (D8)
Area (with t=edge length)	<math>2(1+\sqrt{2})t^2</math> <math> \simeq 4.828427 t^2.</math>
Internal angle (degrees)	135°

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

Contents 1 Regular octagons 2 Uses of octagons 3 See also 4 External links

[edit] 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

<math>A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427 a^2.</math>

In terms of <math>R</math>, (circumradius) the area is

<math>A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427 R^2.</math>

In terms of <math>r</math>, (inradius) the area is

<math>A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085 r^2.</math>

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.
The area may also be found this way:

<math>A=S^2-B^2.</math>

Where <math>S</math> is the span of the octagon, or the second shortest diagonal; and <math>B</math> 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 <math>S</math> the length of a side <math>B</math> is

<math>B = S/(1+\sqrt{2}).</math>

[edit] Uses of octagons

Image:Stop sign MUTCD.svg In many parts of the world, stop signs are in the shape of a regular octagon.	Image:Knopka 8 ugolnik.jpg Push-button	Image:Solonka 8 ugolnik.jpg
Image:Korobka 8 ugolnik.jpg	Image:Korzina 8 ugolnik.jpg	Image:Zont 8 ugolnik.jpg

Image:Octagram.svg An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon.	Image:Great dirhombicosidodecahedron vertfig.png 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.	Image:Octagonal prism.png The octagonal prism contain two octagons.
Image:Tile 488.svg The truncated square tiling has 2 octagons around every vertex.	Image:Great rhombicuboctahedron.png The truncated cuboctahedron has 6 octagons	Image:Octagonal antiprism.png The octagonal antiprism contain two octagons.

[edit] See also

• octagonal number

[edit] External links

• How to find the area of an octagon
• Definition and properties of an octagon With interactive animation
• Eric W. Weisstein, Octagon at MathWorld.

v • d • e Polygons
Triangle · Quadrilateral · Pentagon · Hexagon · Heptagon · Octagon · Enneagon (Nonagon) · Decagon · Hendecagon · Dodecagon · Triskaidecagon · Pentadecagon · Hexadecagon · Heptadecagon · Enneadecagon · Icosagon · Chiliagon · Myriagon

cs:Osmiúhelník

de:Achteck eo:Oklatero es:Octógono fi:Kahdeksankulmio fr:Octogone ht:Oktagòn it:Ottagono ja:八角形 ka:ოქტაგონი ko:팔각형 nl:Achthoek no:Oktogon pl:Ośmiokąt pt:Octógono ru:Восьмиугольник simple:Octagon sr:Осмоугао sv:Oktagon th:รูปแปดเหลี่ยม zh:八边形