Regular octagon  

A regular octagon  
Edges and vertices  8 
Schläfli symbols  {8} t{4} 
Coxeter–Dynkin diagrams  
Symmetry group  Dihedral (D_{8}) 
Area (with a=edge length) 

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}.
Contents
Regular octagons
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 is given by
In terms of , (circumradius) the area is
In terms of , (inradius) the area is
Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be derived as folllows:
where is the span of the octagon, or the second shortest diagonal; and 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 454590 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 , the length of a side is:
The area, is then as above:
Uses of octagons
Derived figures
Petrie polygons
The octagon is the Petrie polygon for these 12 higherdimensional uniform polytopes, shown in these skew orthogonal projections of in A_{7}, B_{4}, and D_{5} Coxeter planes.
A_{7}  7simplex 
Rectified 7simplex 
Birectified 7simplex 
Trirectified 7simplex 

B_{4}  16cell 
Rectified 16cell 
Rectified tesseract 
Tesseract 
D_{5}  Trirectified 5demicube 
Birectified 5demicube 
Rectified 5demicube 
5demicube 
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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