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)	<math>2(1+\sqrt{2})t^2</math> <math> \simeq 4.828427 t^2.</math>
Internal angle (degrees)	135°

In geometry, an octagon is a polygon that has eight sides. Regular octagon is represented by 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 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

<math>A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427 a^2.</math>

In terms of <math>R</math>, (circumradius) the area is

<math>A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427 R^2.</math>

In terms of <math>r</math>, (inradius) the area is

<math>A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085 r^2.</math>

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.
The area may also be found this way:

<math>A=S^2-B^2.</math>

Where <math>S</math> is the span of the octagon, or the second shortest diagonal; and <math>B</math> 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 <math>S</math> the length of a side <math>B</math> is

<math>B = S/(1+\sqrt{2}).</math>

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.	The octagonal prism contain two octagons.
The truncated square tiling has 2 octagons around every vertex.	The truncated cuboctahedron has 6 octagons	The octagonal antiprism contain two octagons.

See also

• octagonal number

External links

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