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Octagon

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Regular octagon
Regular octagon
A regular octagon
Edges and vertices8
Schläfli symbols{8}
t{4}
Coxeter–Dynkin diagramsCDW ringCDW 8CDW dot
CDW ringCDW 4CDW ring
Symmetry groupDihedral (D8)
Area
(with a=edge length)
2(1+\sqrt{2})a^2
 \simeq 4.828427 a^2.
Internal angle
(degrees)
135°

Template:Otheruses

In geometry, an octagon is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

Regular octagons

OctagonConstructionAni

A regular octagon is constructible with compass and straightedge. To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 through 10.

A regular octagon is always an octagon whose sides are all the same length and whose internal angles are all the same size. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080°. The area of a regular octagon of side length a is given by

A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427\,a^2.

In terms of R, (circumradius) the area is

A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427\,R^2.

In terms of r, (inradius) the area is

A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085\,r^2.

Naturally, those last two coefficients bracket the value of pi, the area of the unit circle.

Octagon in square

An octagon inset in a square.

The area can also be derived as folllows:

\,\!A=S^2-a^2,

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 span S, the length of a side a is:

S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a
S=2.414a\,

The area, is then as above:

A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2.

Uses of octagons

Derived figures

Petrie polygons

The octagon is the Petrie polygon for these 12 higher-dimensional uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes.

A7 7-simplex t0
7-simplex
7-simplex t1
Rectified 7-simplex
7-simplex t2
Birectified 7-simplex
7-simplex t3
Trirectified 7-simplex
B4 4-cube t3
16-cell
24-cell t0 B4
Rectified 16-cell
4-cube t1
Rectified tesseract
4-cube t0
Tesseract
D5 5-demicube t3 D5
Trirectified 5-demicube
5-demicube t2 D5
Birectified 5-demicube
5-demicube t1 D5
Rectified 5-demicube
5-demicube t0 D5
5-demicube


See also

External links


ar:ثماني أضلاع ast:Octógonu az:Düzgün səkkizbucaqlı ca:Octàgon cs:Osmiúhelník cy:Octagoneo:Oklaterogl:Octógonoit:Ottagono he:מתומן ka:ოქტაგონი ht:Oktagòn hu:Nyolcszög ms:Oktagon mn:Найман өнцөгт nl:Achthoekno:Oktogon nn:Oktogon pl:Ośmiokąt pt:Octógonosimple:Octagon sk:Osemuholník sl:Osemkotnik sr:Осмоугаоsv:Oktagon ta:எண்கோணம் th:รูปแปดเหลี่ยม

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