The regular octagon is the regular polygon with eight sides, as illustrated above.
The inradius 
, circumradius 
, and area 
of the regular octagon can be computed directly from
the formulas for a general regular polygon with
side length 
and 
sides as
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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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The vertex angle 
, central angle 
, and exterior angle 
are given by
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(7)
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(8)
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(9)
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The octagon with alternate sides parallel to the 
- or 
-axes as illustrated above is the shape used for traffic stop
signs in the United States.
An octagon with inradius 1 in this orientation can be specified by the inequalities
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(10)
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The octagon may also be orientated so that its vertices lie along the 
- and 
-axes and at 
degree angles to them, as illustrated above. An octagon
with circumradius 
in this orientation can be specified by the inequality
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(11)
|
The Castel del Monte in Southern Italy consists of a central octagonal core with an inner octagonal courtyard. This, in turn, is surrounded by eight perfectly octagonal towers. Other examples of octagonal structures in art and architecture include octagonal Roman mosaics and tiling patterns in the Alhambra Palace in Spain, Jerusalem's Dome of the Rock, and the Church of San Vitale in Ravenna (Gotze 1998).
