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

• octagram
• octagonal number
• octagon house
• bumper pool
• Octagon Worldwide

External links

• How to find the area of an octagon
• Definition and properties of an octagon With interactive animation