Octagon

Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8} t{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D8)
Area (with a=edge length)	$2\left(1+{\sqrt {2}}\right)a^{2}$ $\simeq 4.828a^{2}$
Internal angle (degrees)	135°

Template:Otheruses

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 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

$A=2\cot \left({\frac {\pi }{8}}\right)a^{2}=2\left(1+{\sqrt {2}}\right)a^{2}\simeq 4.828a^{2}$

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

$A=4\sin \left({\frac {\pi }{4}}\right)R^{2}=2{\sqrt {2}}R^{2}\simeq 2.828R^{2}$

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

$A=8\tan \left({\frac {\pi }{8}}\right)r^{2}=8\left({\sqrt {2}}-1\right)r^{2}\simeq 3.313r^{2}$

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.

An octagon inset in a square.

The area can also be derived as folllows:

$A=S^{2}-a^{2}$

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:

$S={\frac {a}{\sqrt {2}}}+a+{\frac {a}{\sqrt {2}}}=\left(1+{\sqrt {2}}\right)a$

$S=2.414a$

The area, is then as above:

$A={\bigl (}(1+{\sqrt {2}})a{\bigr )}^{2}-a^{2}=2\left(1+{\sqrt {2}}\right)a^{2}$

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.

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.

Petrie polygons

The octagon is the Petrie polygon for these 12 higher-dimensional uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

A7	7-simplex	Rectified 7-simplex	Birectified 7-simplex	Trirectified 7-simplex
B4	16-cell	Rectified 16-cell	Rectified tesseract	Tesseract
D5	Trirectified 5-demicube	Birectified 5-demicube	Rectified 5-demicube	5-demicube

See also

• Octagram
• Octagonal number
• Octagon house
• Bumper pool
• Leonardo da Vinci

External links

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

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