In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, an
octagon (from the
GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
okto, eight) is a
polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
that has
eight sides. A regular octagon is represented by the
Schläfli symbol {8}.
Regular octagon
A
regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
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 symmetryGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
of order 8.
The internal
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
at each vertex of a regular octagon is 135
°A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
and the sum of all the internal angles is 1080° (as for any octagon).
The area of a regular octagon of side length
a is given by
In terms of
(circumradius), the area is
In terms of
(
inradiusrightthumbAn inscribed triangle of a circleIn geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "Figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about...
), the area is
These last two coefficients bracket the value of
pi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, the area of the
unit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
.
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 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 length of a side
, the span
is:
(approximately)
The area is then as above:
Another simple formula for the area is
where d is the distance between parallel sides
(the same as span S in the diagram).
Construction
A regular octagon is
constructibleIn mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....
with
compass and straightedgeCompassandstraightedge or rulerandcompass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
.
Standard coordinates
The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:
 (±1, ±(1+√2))
 (±(1+√2), ±1).
Petrie polygons
The octagon is the
Petrie polygonIn geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...
for these 12 higher dimensional
uniform polytopeA uniform polytope is a vertextransitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
s, shown in these skew orthogonal projections of in A
_{7}, B
_{4}, and D
_{5} Coxeter planes.
A_{7} 
7simplex 
Rectified 7simplexIn sevendimensional geometry, a rectified 7simplex is a convex uniform 7polytope, being a rectification of the regular 7simplex.There are four unique degrees of rectifications, including the zeroth, the 7simplex itself. Vertices of the rectified 7simplex are located at the edgecenters of the...

Birectified 7simplex 
Trirectified 7simplex 
B_{4} 
16cellIn four dimensional geometry, a 16cell or hexadecachoron is a regular convex 4polytope. It is one of the six regular convex 4polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid19th century....

Rectified 16cell 
Rectified tesseractIn geometry, the rectified tesseract, or rectified 8cell is a uniform polychoron bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra....

TesseractIn geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

D_{5} 
Trirectified demipenteract 
Birectified demipenteract 
Rectified demipenteract 
DemipenteractIn five dimensional geometry, a demipenteract or 5demicube is a semiregular 5polytope, constructed from a 5hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...

See also
 Octagram
In geometry, an octagram is an eightsided star polygon. Geometry :In general, an octagram is any selfintersecting octagon ....
 Octagonal number
 Octagon house
Octagon houses were a unique house style briefly popular in the 1850s in the United States and Canada. They are characterised by an octagonal plan, and often feature a flat roof and a veranda all round...
 Oktogon
Oktogon is one of Pest's major intersections, located at the junction of the Grand Boulevard and Andrássy Avenue in Budapest, Hungary...
, a major intersection in BudapestBudapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
, HungaryHungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
 Bumper pool
Bumper pool is a pocket billards game played on an octagonal or rectangular table fitted with an array of fixed cushioned obstacles, called bumpers, at the center of its surface. Table :...
 Rub el Hizb
The Rub el Hizb is a Muslim symbol, represented as two overlapping squares, which is found on a number of emblems and flags. In Arabic, Rubʻ means "one fourth, quarter", while Hizb means a group or party...
(also known as Al Quds Star and as Octa Star)
 Smoothed octagon
The smoothed octagon is a geometrical construction conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes...
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