An octagon is an eight-sided polygon. 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

			(1)
			(2)
			(3)

The vertex angle , central angle , and exterior angle are given by

			(4)
			(5)
			(6)

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

(7)

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

(8)

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).

Gotze, H. Castel del Monte, Geometric Marvel of the Middle Ages. Munich, Germany: Prestel, 1998.

Signalfan. "History of the Stop Sign in America." http://signalfan.freeservers.com/road%20signs/stopsign.htm.

CITE THIS AS:

Weisstein, Eric W. "Octagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html