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

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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

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

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

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

The area may also be found this way:

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


The area, then, is

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