TOPICS
Search

Regular Octagon


RegularOctagon

The regular octagon is the regular polygon with eight sides, as illustrated above.

OctagonAngles

The inradius r, circumradius R, and area A of the regular octagon can be computed directly from the formulas for a general regular polygon with side length a and n=8 sides as

r=1/2acot(pi/8)
(1)
=1/2(1+sqrt(2))a
(2)
R=1/2acsc(pi/8)
(3)
=1/2sqrt(4+2sqrt(2))a
(4)
A=1/4na^2cot(pi/8)
(5)
=2(1+sqrt(2))a^2.
(6)

The vertex angle alpha, central angle theta, and exterior angle beta are given by

theta=1/4pi
(7)
alpha=3/4pi
(8)
beta=5/4pi.
(9)
American stop sign

The octagon with alternate sides parallel to the x- or y-axes as illustrated above is the shape used for traffic stop signs in the United States.

Octagon1

An octagon with inradius 1 in this orientation can be specified by the inequalities

 -1<(x-y)/(sqrt(2))<1 ^ -1<(x+y)/(sqrt(2))<1 ^ -1<x<1 ^  
 -1<y<1.
(10)
Octagon2

The octagon may also be orientated so that its vertices lie along the x- and y-axes and at 45 degrees degree angles to them, as illustrated above. An octagon with circumradius (sqrt(2)-1)/2 in this orientation can be specified by the inequality

 2(|x|+|y|)+sqrt(2)(|x-y|+|x+y|)<1.
(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).


See also

Octagon, Octahedron, Octagram, Polygon, Regular Polygon, Smoothed Octagon, Star of Lakshmi, Trigonometry Angles--Pi/8

Explore with Wolfram|Alpha

References

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. "Regular Octagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularOctagon.html

Subject classifications