Math Wiki
Regular octagon
Regular octagon.svg
A regular octagon
Edges and vertices 8
Schläfli symbols {8}
Coxeter–Dynkin diagrams CDW ring.pngCDW 8.pngCDW dot.png
CDW ring.pngCDW 4.pngCDW ring.png
Symmetry group Dihedral (D8)
(with a=edge length)

Internal angle


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

Regular octagons

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 is given by

In terms of , (circumradius) the area is

In terms of , (inradius) the area is

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

An octagon inset in a square.

The area can also be derived as folllows:

where is the span of the octagon, or the second shortest diagonal; and 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 , the length of a side is:

The area, is then as above:

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

See also

  • Octagram
  • Octagonal number
  • Octagon house
  • Bumper pool
  • Leonardo da Vinci

External links

ar:ثماني أضلاع ast:Octógonu az:Düzgün səkkizbucaqlı ca:Octàgon cs:Osmiúhelník cy:Octagon eo:Oklatero gl:Octógono it:Ottagono he:מתומן ka:ოქტაგონი ht:Oktagòn hu:Nyolcszög ms:Oktagon mn:Найман өнцөгт nl:Achthoek no:Oktogon nn:Oktogon pl:Ośmiokąt pt:Octógono simple:Octagon sk:Osemuholník sl:Osemkotnik sr:Осмоугао sv:Oktagon ta:எண்கோணம் th:รูปแปดเหลี่ยม