Octagon

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Regular octagon

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)		 2(1+\sqrt{2})t^2
\simeq 4.828427 t^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}.

Contents

[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 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 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 may also be found this way:

A = S2 − B2.

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.This is the easier way to find such an area. However, one of the most accurate ways is as follows: Take the coefficient of the square root of one of the sides and multiply by three! This is proved by the fact that coefficient of pi, proved by Archidimes, is equal to two times the sin of any given triangle as long as all equal sides add up to the factorial of the scalene ones, especially those of such a square. Seeing as an octagon has three of these triangles in the interior of a convex Given the span S the length of a side B is

B = S/(1+\sqrt{2})./ ,S1 + S2 + S3 + S4 + S5 + S6 +S7 + S8

[edit] 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.

[edit] See also

[edit] External links

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Polygons