Definitions

Octagon

Octagon

[ok-tuh-gon, -guhn]
Regular octagon

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

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

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

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 = 2sqrt{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=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

S=frac{B}{sqrt{2}}+B+frac{B}{sqrt{2}}=(1+sqrt{2})B
S=2.414B

The area, then, is

A=((1+sqrt{2})B)^2-B^2=2(1+sqrt{2})B^2.

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.

See also

External links

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