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{{Even polygon db|Even polygon stat table|p8}}

In [[geometry]], an '''octagon''' (from the [[Ancient Greek|Greek]] ὀκτάγωνον ''oktágōnon'', "eight angles") is a polygon that has [[8 (number)|eight]] sides. 

==Regular octagon==
A [[regular polygon|regular]] 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]] 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)|°]] and the sum of all the internal angles of any octagon is 1080° (as with all polygons, the external angles total 360°). 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 \simeq 4.828427125\,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 \simeq 2.828427\,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 \simeq 3.3137085\,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 derived as follows:
:<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 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 ''a'', the span ''S'' is:
:<math>S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a</math>
:<math>S \approx 2.414a</math>

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

Expressed in terms of the span, area is:
:<math>A=2(\sqrt{2}-1)S^2</math>
:<math>A \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, as well as being ''e=a/sqrt{2}'', may be calculated as:
:<math>\,\!e=(S-a)/2</math>

==Construction and elementary properties==
<!-- I'd really like to keep the movie, for this simpler procedure, but I don't know how... -->
A regular octagon may be constructed as follows:
#Draw a circle and a diameter AOB, where O is the center and A,B are points on the circumference.
#Draw another diameter COD, perpendicular to AOB.
#(Note in passing that A,B,C,D are vertices of a square).
#Draw the bisectors of the right angles AOC and BOC, making two more diameters EOF and GOH.
#A,B,C,D,E,F,G,H are the vertices of the octagon.

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 8 isosceles triangles, leading to the result:
Area = 2*a*a*(sqrt(2) + 1)   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+√2)) 
*(±(1+√2), ±1).

==Uses of octagons==
[[File:Dehio 10 Dome of the Rock Floor plan.jpg|thumb|The octagonal floor plan, Dome of the Rock.]]
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), [[Castel del Monte, Apulia|Castel del Monte]] (Apulia), [[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]].

===Other uses===
<gallery perrow=4>
File:zont 8 ugolnik.jpg|Umbrellas 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|[[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.
</gallery>

==Derived figures==
<gallery>
File:Octagram.svg|An [[Octagram|eight-sided star]], called an ''octagram'', with Schläfli symbol {8/3} is contained with a regular octagon.
File:Tiling Semiregular 4-8-8 Truncated Square.svg|The [[truncated square tiling]] has 2 octagons around every vertex.
File:Octagonal prism.png|An [[octagonal prism]] contains two octagonal faces.
File:Octagonal antiprism.png|An [[octagonal antiprism]] contains two octagonal faces.
File:Great rhombicuboctahedron.png|The [[truncated cuboctahedron]] contains 6 octagonal faces.
</gallery>

=== Related polytopes ===
The ''octagon'', as a [[Truncation (geometry)|truncated]] [[square]], is first in a sequence of truncated [[hypercube]]s:
{{Truncated hypercube polytopes}}

===Petrie polygons===
The octagon is the [[Petrie polygon]] for these 12 higher dimensional [[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>
|[[File:7-simplex t0.svg|150px]]<br>[[7-simplex]]
|[[File:7-simplex t1.svg|150px]]<br>[[Rectified 7-simplex]]
|[[File:7-simplex t2.svg|150px]]<br>[[Birectified 7-simplex]]
|[[File:7-simplex t3.svg|150px]]<br>[[Trirectified 7-simplex]]
|- align=center
!B<sub>4</sub>
|[[File:4-cube t3.svg|150px]]<br>[[16-cell]]
|[[File:4-cube t2.svg|150px]]<br>[[Rectified 16-cell]]
|[[File:4-cube t1.svg|150px]]<br>[[Rectified tesseract]]
|[[File:4-cube t0.svg|150px]]<br>[[Tesseract]]
|- align=center
!D<sub>5</sub>
|[[File:5-demicube t3 D5.svg|150px]]<br>[[5-orthoplex|Trirectified 5-demicube]]
|[[File:5-demicube t2 D5.svg|150px]]<br>[[Rectified 5-orthoplex|Birectified 5-demicube]]
|[[File:5-demicube t1 D5.svg|150px]]<br>[[Rectified 5-demicube]]
|[[File:5-demicube t0 D5.svg|150px]]<br>[[5-demicube]]
|}

==See also==
*[[Bumper pool]]
*[[Octagon house]]
*[[Octagonal number]]
*[[Octagram]]
*[[Oktogon]], a major intersection in [[Budapest]], [[Hungary]]
*[[Rub el Hizb]] (also known as Al Quds Star and as Octa Star)
*[[Smoothed octagon]]

==References==
{{reflist}}

==External links==
*[http://rechneronline.de/pi/octagon.php Octagon Calculator]
*[http://www.mathopenref.com/octagon.html Definition and properties of an octagon] With interactive animation
*{{MathWorld|title=Octagon|urlname=Octagon}}

{{Polygons}}

[[Category:Polygons]]
[[Category:Elementary shapes]]