Octagon

Octagon

Overview
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an octagon (from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European 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 polygon
Polygon
In 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}.

A regular
Regular polygon
A 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 symmetry
Rotational symmetry
Generally 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 angle
Angle
In 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°
Degree (angle)
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 (inradius
Inscribe
right|thumb|An 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
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 circle
Unit circle
In 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.
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Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an octagon (from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European 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 polygon
Polygon
In 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 regular
Regular polygon
A 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 symmetry
Rotational symmetry
Generally 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 angle
Angle
In 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°
Degree (angle)
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 (inradius
Inscribe
right|thumb|An 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
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 circle
Unit circle
In 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 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 , 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 constructible
Constructible polygon
In 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 straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass 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 polygon
Petrie polygon
In 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 polytope
Uniform polytope
A uniform polytope is a vertex-transitive 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 A7, B4, and D5 Coxeter planes.
A7
7-simplex

Rectified 7-simplex
Rectified 7-simplex
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...


Birectified 7-simplex

Trirectified 7-simplex
B4
16-cell
16-cell
In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....


Rectified 16-cell

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


Tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional 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...

D5
Trirectified demipenteract

Birectified demipenteract

Rectified demipenteract

Demipenteract
Demipenteract
In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...


See also

  • Octagram
    Octagram
    In geometry, an octagram is an eight-sided star polygon.- Geometry :In general, an octagram is any self-intersecting octagon ....

  • Octagonal number
  • Octagon house
    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
    ‎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 Budapest
    Budapest
    Budapest 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...

    , Hungary
    Hungary
    Hungary , 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
    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
    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
    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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