Octagon
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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° 
In geometry, an octagon is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.
Regular octagonsEdit
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 a 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 S is the span of the octagon, or the second shortest diagonal; and a 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 Edit
Derived figuresEdit
Petrie polygonsEdit
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.
See alsoEdit
External linksEdit
 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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