Editing Octagon (section)

==Properties==

[[File:A generalization van Aubel theorem.svg|thumb|left|400px|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|508px]]
{{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
|}