Octagon

Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8} t{4}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D₈)
Area (with a=edge length)	$ 2\left(1+\sqrt2\right)a^2 $ $ \simeq4.828a^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\left(\frac{\pi}{8}\right)a^2=2\left(1+\sqrt2\right)a^2\simeq4.828a^2 $

In terms of $ R $ , (circumradius) the area is

$ A=4\sin\left(\frac{\pi}{4}\right)R^2=2\sqrt2R^2\simeq2.828R^2 $

In terms of $ r $ , (inradius) the area is

$ A=8\tan\left(\frac{\pi}{8}\right)r^2=8\left(\sqrt2-1\right)r^2\simeq3.313r^2 $

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

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}{\sqrt2}+a+\frac{a}{\sqrt2}=\left(1+\sqrt2\right)a $

$ S=2.414a $

The area, is then as above:

$ A=\bigl((1+\sqrt2)a\bigr)^2-a^2=2\left(1+\sqrt2\right)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	Rectified 7-simplex	Birectified 7-simplex	Trirectified 7-simplex
B₄	16-cell	Rectified 16-cell	Rectified tesseract	Tesseract
D₅	Trirectified 5-demicube	Birectified 5-demicube	Rectified 5-demicube	5-demicube