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Regular octagon  

A regular octagon  
Type  Regular polygon 
Edges and vertices  8 
Schläfli symbol  {8} t{4} 
Coxeter diagram  
Symmetry group  D_{8}, order 2×8 
Internal angle (degrees)  135° 
Dual polygon  self 
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
Regular octagon  

A regular octagon  
Type  Regular polygon 
Edges and vertices  8 
Schläfli symbol  {8} t{4} 
Coxeter diagram  
Symmetry group  D_{8}, order 2×8 
Internal angle (degrees)  135° 
Dual polygon  self 
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is a polygon that has eight sides.
A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles of any octagon is 1080° (as with all polygons, the external angles total 360°). The area of a regular octagon of side length a is given by
In terms of the circumradius R, the area is
In terms of the apothem r (see also inscribed figure), the area is
These last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be derived as follows:
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 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 length of a side a, the span S is:
The area is then as above:
Expressed in terms of the span, area is:
Another simple formula for the area is
More often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above:
The two end lengths e on each side, as well as being e=a/sqrt{2}, may be calculated as:
A regular octagon may be constructed as follows:
Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of 8 isosceles triangles, leading to the result:
for an octagon of side a.
The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:
The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna), Castel del Monte (Apulia), Florence Baptistery, Zum Friedefürsten church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral.
Umbrellas often have an octagonal outline.
The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.
Janggi uses octagonal pieces.
Japanese lottery machines often have octagonal shape.
Stop sign used in Englishspeaking countries, as well as in most European countries
Famous octagonal gold cup from the Belitung shipwreck
Classes at Shimer College are traditionally held around octagonal tables
The Labyrinth of the Reims Cathedral with a quasioctagonal shape.
A vertextransitive (isogonal) octagon is construct in four mirrors can alternate long and short edges. An edgetransitive octagon (isotoxal) is constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon.
Isogonal  Isotoxal 
The truncated square tiling has 2 octagons around every vertex.
An octagonal prism contains two octagonal faces.
An octagonal antiprism contains two octagonal faces.
The truncated cuboctahedron contains 6 octagonal faces.
The octagon, as a truncated square, is first in a sequence of truncated hypercubes:
...  
Octagon  Truncated cube  Truncated tesseract  Truncated 5cube  Truncated 6cube  Truncated 7cube  Truncated 8cube  
As an expanded square, it is also first in a sequence of expanded hypercubes:
...  
Octagon  Rhombicuboctahedron  Runcinated tesseract  Stericated 5cube  Pentellated 6cube  Hexicated 7cube  Heptellated 8cube  
The octagon is the Petrie polygon for these higherdimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A_{7}, B_{4}, and D_{5} Coxeter planes.
A_{7}  D_{5}  B_{4}  

7simplex  5demicube  16cell  Tesseract 
