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{{Short description|Polygon shape with eight sides}}
{{Redirect|Octagonal|other uses|Octagon (disambiguation)|and|Octagonal (disambiguation)}}
{{Regular polygon db|Regular polygon stat table|p8}}

In [[geometry]], an '''octagon''' ({{etymology|grc|''ὀκτάγωνον'' ({{grc-transl|ὀκτάγωνον}})|eight angles}}) is an eight-sided [[polygon]] or 8-gon.

A ''[[regular polygon|regular]] octagon'' has [[Schläfli symbol]] {8}<ref>{{citation|title=Polyhedron Models|first=Magnus J.|last=Wenninger|publisher=Cambridge University Press|year=1974|page=9|isbn=9780521098595|url=https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9}}.</ref> and can also be constructed as a quasiregular [[Truncation (geometry)|truncated]] [[square]], t{4}, which alternates two types of edges. A truncated octagon, t{8} is a [[hexadecagon]], {16}. A 3D analog of the octagon can be the [[rhombicuboctahedron]] with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.

==Properties==

[[File:A generalization van Aubel theorem.svg|thumb|left|The diagonals of the green quadrilateral are equal in length and at right angles to each other]]

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.

If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both [[equidiagonal quadrilateral|equidiagonal]] and [[orthodiagonal quadrilateral|orthodiagonal]] (that is, whose diagonals are equal in length and at right angles to each other).<ref name=Oai>Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", ''Forum Geometricorum'' 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html {{Webarchive|url=https://web.archive.org/web/20150705033424/http://forumgeom.fau.edu/FG2015volume15/FG201509index.html |date=2015-07-05 }}</ref>{{rp|Prop. 9}}

The [[midpoint polygon|midpoint octagon]] of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.<ref name=Oai/>{{rp|Prop. 10}}

===Regularity===
A [[regular polygon|regular]] octagon is a closed [[Shape|figure]] with sides of the same length and internal angles of the same size. It has eight lines of [[reflective symmetry]] and [[rotational symmetry]] of order 8. A regular octagon is represented by the [[Schläfli symbol]] {8}.
The internal [[angle]] at each vertex of a regular octagon is 135[[degree (angle)|°]] (<math>\scriptstyle \frac{3\pi}{4}</math> [[radian]]s). The [[central angle]] is 45° (<math>\scriptstyle \frac{\pi}{4}</math> radians).

=== Area ===
The area of a regular octagon of side length ''a'' is given by
:<math>A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \approx 4.828\,a^2.</math>

In terms of the [[Circumscribed circle|circumradius]] ''R'', the area is
:<math>A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \approx 2.828\,R^2.</math>

In terms of the [[apothem]] ''r'' (see also [[inscribed figure]]), the area is
:<math>A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \approx 3.314\,r^2.</math>

These last two [[coefficients]] bracket the value of [[pi]], the area of the [[unit circle]].
[[File:Octagon in square.svg|frame|The [[area]] of a [[Regular polygon|regular]] octagon can be computed as a [[Truncation (geometry)|truncated]] [[Square (geometry)|square]].]]

The area can also be expressed as
:<math>\,\!A=S^2-a^2,</math>

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 overlap with the four sides of the square) and then takes the corner triangles (these are [[Special right triangles#45–45–90 triangle|45–45–90 triangles]]) and places 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 ''a'', the span ''S'' is
:<math>S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a \approx 2.414a.</math>
The span, then, is equal to the ''[[silver ratio]]'' times the side, a.

The area is then as above:
:<math>A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2 \approx 4.828a^2.</math>

Expressed in terms of the span, the area is
:<math>A=2(\sqrt{2}-1)S^2 \approx 0.828S^2.</math>

Another simple formula for the area is 
:<math>\ A=2aS.</math>

More often the span ''S'' is known, and the length of the sides, ''a'', is to be determined, as when cutting a square piece of material into a regular octagon. From the above,
:<math>a \approx S/2.414.</math>

The two end lengths ''e'' on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being <math>e=a/\sqrt{2},</math> may be calculated as
:<math>\,\!e=(S-a)/2.</math>

===Circumradius and inradius===

The [[circumradius]] of the regular octagon in terms of the side length ''a''  is<ref>Weisstein, Eric. "Octagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html</ref>

:<math>R=\left(\frac{\sqrt{4+2\sqrt{2}}}{2}\right)a \approx 1.307 a,</math>

and the [[inradius]] is

:<math>r=\left(\frac{1+\sqrt{2}}{2}\right)a \approx 1.207 a.</math>
(that is one-half the ''[[silver ratio]]'' times the side, ''a'', or one-half the span, ''S'')

The inradius can be calculated from the circumradius as
:<math>r = R \cos \frac{\pi}{8}</math>

===Diagonality===
The regular octagon, in terms of the side length ''a'', has three different types of [[diagonal]]s:

*Short diagonal;
*Medium diagonal (also called span or height), which is twice the length of the inradius;
*Long diagonal, which is twice the length of the circumradius.

The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:<ref>{{citation|title=A Panoply of Polygons|volume=58|series=Dolciani Mathematical Expositions|first1=Claudi|last1=Alsina|first2=Roger B.|last2=Nelsen|publisher=American Mathematical Society|year=2023|isbn=9781470471842|page=124|url=https://books.google.com/books?id=LqatEAAAQBAJ&pg=PA124}}</ref>

*Short diagonal: <math>a\sqrt{2+\sqrt2}</math> ;
*Medium diagonal: <math>(1+\sqrt2)a</math> ; (''[[silver ratio]]'' times a) 
*Long diagonal: <math>a\sqrt{4 + 2\sqrt2}</math> .

===Construction===
[[File:8-folding.svg|thumb|left|building a regular octagon by folding a sheet of paper]]
<!-- I'd really like to keep the movie, for this simpler procedure, but I don't know how... -->
{{clear}}
A regular octagon at a given circumcircle may be constructed as follows:
#Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
#Draw another diameter GOC, perpendicular to AOE.
#(Note in passing that A,C,E,G are vertices of a square).
#Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
#A,B,C,D,E,F,G,H are the vertices of the octagon.
{{multiple image
| align = left
| image1 = 01-Octagon.svg
| width1 = 482 
| alt1 = 
| caption1 = Octagon at a given circumcircle
| image2 = 01-Achteck-Seite-gegeben Animation.gif
| width2 = 345
| alt2 = 
| caption2 = Octagon at a given side length, animation<br />
(The construction is very similar to that of [[Hexadecagon#Construction|hexadecagon at a given side length]].)
| footer = 
}}
{{clear}}

A regular octagon can be constructed using a [[straightedge]] and a [[Compass (drawing tool)|compass]], as 8 = 2<sup>3</sup>, a [[power of two]]:

[[File:Regular Octagon Inscribed in a Circle.gif|left|frameless]]
{{clear}}

[[File:Meccano octagon.svg|thumb|Meccano octagon construction.]]
The regular octagon can be constructed with [[meccano]] bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required.
{{clear}}

Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices.  Its area can thus be computed as the sum of eight isosceles triangles, leading to the result:

:<math>\text{Area} = 2 a^2 (\sqrt{2} + 1)</math>

for an octagon of side ''a''.

===Standard coordinates===
The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:
*(±1, ±(1+{{radic|2}})) 
*(±(1+{{radic|2}}), ±1).

===Dissectibility===
{| class=wikitable align=right
![[8-cube]] projection
!colspan=2|24 rhomb dissection
|- align=center
|[[File:8-cube t0 A7.svg|160px]]
|[[File:8-gon rhombic dissection-size2.svg|160px]]<BR>Regular
|[[File:Isotoxal 8-gon rhombic dissection-size2.svg|160px]]<BR>Isotoxal
|-
|[[File:8-gon rhombic dissection2-size2.svg|160px]]
|[[File:8-gon rhombic dissection3-size2.svg|160px]]
|}

[[Coxeter]] states that every [[zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.<ref>[[Coxeter]], Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ''regular octagon'', ''m''=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a [[Petrie polygon]] projection plane of the [[tesseract]]. The list {{OEIS|1=A006245}} defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the [[Ammann–Beenker tiling]]s.
{| class=wikitable
|+ Regular octagon dissected
|- align=center valign=top
|[[File:4-cube_t0.svg|160px]]<BR>[[Tesseract]]
|[[File:Dissected octagon.svg|160px]]<BR>4 rhombs and 2 squares
|}

== Skew ==
[[File:Skew polygon in square antiprism.svg|thumb|A regular skew octagon seen as edges of a [[square antiprism]], symmetry D<sub>4d</sub>, [2<sup>+</sup>,8], (2*4), order 16.]]
A '''skew octagon''' is a [[skew polygon]] with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A ''skew zig-zag octagon'' has vertices alternating between two parallel planes.

A '''regular skew octagon''' is [[vertex-transitive]] with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a [[square antiprism]] with the same D<sub>4d</sub>, [2<sup>+</sup>,8] symmetry, order 16.

===Petrie polygons===
The regular skew octagon is the [[Petrie polygon]] for these higher-dimensional regular and [[uniform polytope]]s, shown in these skew [[orthogonal projection]]s of in A<sub>7</sub>, B<sub>4</sub>, and D<sub>5</sub> [[Coxeter plane]]s.
{| class=wikitable width=600
|- align=center
!A<sub>7</sub>
!D<sub>5</sub>
!colspan=2|B<sub>4</sub>
|- align=center
|[[File:7-simplex t0.svg|150px]]<br>[[7-simplex]]
|[[File:5-demicube t0 D5.svg|150px]]<br>[[5-demicube]]
|[[File:4-cube t3.svg|150px]]<br>[[16-cell]]
|[[File:4-cube t0.svg|150px]]<br>[[Tesseract]]
|}

==Symmetry==
{| class=wikitable width=380 align=right
|+ Symmetry
|- valign=top
|[[File:Regular_octagon_symmetries.png|220px]]
|The eleven symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.
|}
The ''regular octagon'' has Dih<sub>8</sub> symmetry, order 16. There are three dihedral subgroups: Dih<sub>4</sub>, Dih<sub>2</sub>, and Dih<sub>1</sub>, and four [[cyclic group|cyclic subgroups]]: Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>, and Z<sub>1</sub>, the last implying no symmetry.
{| class=wikitable align=left
|+ Example octagons by symmetry
|- align=center
!colspan=3|[[File:Octagon_r16_symmetry.png|60px]]<BR>r16
|-
![[File:Octagon_d8_symmetry.png|60px]]<BR>d8
![[File:Octagon_g8_symmetry.png|60px]]<BR>g8
![[File:Octagon_p8_symmetry.png|60px]]<BR>p8
|-
![[File:Octagon_d4_symmetry.png|60px]]<BR>d4
![[File:Octagon_g4_symmetry.png|60px]]<BR>g4
![[File:Octagon_p4_symmetry.png|60px]]<BR>p4
|-
![[File:Octagon_d2_symmetry.png|60px]]<BR>d2
![[File:Octagon_g2_symmetry.png|60px]]<BR>g2
![[File:Octagon_p2_symmetry.png|60px]]<BR>p2
|- align=center
!colspan=3|[[File:Octagon_a1_symmetry.png|60px]]<BR>a1
|}

On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as '''r16'''.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref> The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars) Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders. Full symmetry of the regular form is '''r16''' and no symmetry is labeled '''a1'''.

The most common high symmetry octagons are '''p8''', an [[isogonal figure|isogonal]] octagon constructed by four mirrors can alternate long and short edges, and '''d8''', an [[isotoxal figure|isotoxal]] octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[dual polygon|duals]] of each other and have half the symmetry order of the regular octagon.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g8''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.
{{Clear}}

==Use==
[[File:Dehio 10 Dome of the Rock Floor plan.jpg|thumb|The octagonal floor plan, [[Dome of the Rock]], [[Jerusalem]].]]
The octagonal shape is used as a design element in architecture. The [[Dome of the Rock]] has a characteristic octagonal plan. The [[Tower of the Winds]] in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as [[St. George's Cathedral, Addis Ababa]], [[Basilica of San Vitale]] (in Ravenna, Italia), [[Castel del Monte, Apulia|Castel del Monte]] (Apulia, Italia), [[Florence Baptistery]], [[Zum Friedefürsten Church]] (Germany) and a number of [[octagonal churches in Norway]]. The central space in the [[Aachen Cathedral]], the Carolingian [[Palatine Chapel, Aachen|Palatine Chapel]], has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal [[apse]] of [[Nidaros Cathedral]].

Architects such as [[John Andrews (architect)|John Andrews]] have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in the [[Intelsat Headquarters]] of Washington or [[Callam Offices]] in Canberra.

<gallery perrow="4">
File:Zont 8 ugolnik.jpg|[[Umbrella]]s often have an octagonal outline.
File:Afghancarpet1.jpg|The famous [[Bukhara rug]] design incorporates an octagonal "elephant's foot" motif.
File:Eixample.svg|The street & block layout of [[Barcelona]]'s [[Eixample]] district is based on non-regular octagons
File:Janggipieces.jpg|[[Janggi]] uses octagonal pieces.
File:Revolving lottery machine,kaitenshiki-cyusenki,japan.JPG|Japanese [[lottery machine]]s often have octagonal shape.
File:MUTCD R1-1.svg|A [[Stop sign]] used in [[English language|English]]-speaking countries, as well as in most [[European countries]]
File:Bagua-name-earlier.svg|The trigrams of the [[Taoism|Taoist]] ''[[bagua]]'' are often arranged octagonally
File:Octagonal footed gold cup from the Belitung shipwreck, ArtScience Museum, Singapore - 20110618-01.jpg|Famous octagonal gold cup from the [[Belitung shipwreck]]
File:Shimer College class 1995 octagonal table.jpg|Classes at [[Shimer College]] are traditionally held around octagonal tables
File:Labyrinthe de la cathédrale de Reims.svg|The [[Labyrinth of the Reims Cathedral]] with a quasi-octagonal shape.
File:GameCube Analog Stick.jpg|The movement of the [[analog stick]](s) of the [[Nintendo 64 controller]], the [[GameCube controller]], the [[Wii Nunchuk]] and the [[Classic Controller]] is bounded by an octagonal frame, helping the user aim the stick in [[cardinal direction]]s while still allowing circular freedom.
File:ALaRonde OctagonChair2 January2024 NT CCBYSA open.jpg|Chair from [[A la Ronde]], with octagonal seats and backs (set of eight)
</gallery>

==Derived figures==

<gallery>
File:Tiling Semiregular 4-8-8 Truncated Square.svg|The [[truncated square tiling]] has 2 octagons around every vertex.<br>{{CDD|node_1|4|node_1|4|node_1}}
File:Octagonal prism.png|An [[octagonal prism]] contains two octagonal faces.<br>{{CDD|node_1|4|node_1|2|node_1}}
File:Octagonal antiprism.png|An [[octagonal antiprism]] contains two octagonal faces.<br>{{CDD|node_h|8|node_h|2x|node_h}}
File:Great rhombicuboctahedron.png|The [[truncated cuboctahedron]] contains 6 octagonal faces.<br>{{CDD|node_1|4|node_1|3|node_1}}
File:Omnitruncated cubic honeycomb2.png|The [[omnitruncated cubic honeycomb]]<br>{{CDD|node_1|4|node_1|3|node_1|4|node_1}}
</gallery>

=== Related polytopes ===
The ''octagon'', as a [[Truncation (geometry)|truncated]] [[square]], is first in a sequence of truncated [[hypercube]]s:
{{Truncated hypercube polytopes}}
As an [[expansion (geometry)|expanded]] square, it is also first in a sequence of expanded hypercubes:
{{Expanded hypercube polytopes}}

==See also==
*[[Bumper pool]]
*[[Hansen's small octagon]]
*[[Octagon house]]
*[[Octagonal number]]
*[[Octagram]]
*[[Octahedron]], 3D shape with eight faces.
*[[Oktogon (intersection)|Oktogon]], a major intersection in [[Budapest]], [[Hungary]]
*[[Rub el Hizb]] (also known as Al Quds Star and as Octa Star), a common motif in [[Islamic architecture]]
*[[Smoothed octagon]]

==References==
{{Reflist}}

==External links==
{{Wiktionary}}

*[http://rechneronline.de/pi/octagon.php Octagon Calculator]
*[http://www.mathopenref.com/octagon.html Definition and properties of an octagon] With interactive animation

{{Polygons}}

[[Category:8 (number)]]
[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]
[[Category:Elementary shapes]]