Octagon

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

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 [ irregular and regular

[regular polygon|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.

An octagon inset in a square.

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
File:Octagon diagram for area derivation length comparison.jpg

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

$S=2.414B$

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

• octagram
• octagonal number
• octagon house
• bumper pool

External links

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