Octagon

Regular octagon
A regular octagon
Edges and vertices	8
Schläfli symbols	{8}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D₈)
Area (with a=edge length)	$2(1+\sqrt{2})a^2$ $\simeq 4.828427 a^2$
Internal angle (degrees)	135°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

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

1 Regular octagons
2 Standard coordinates
3 Uses of octagons
4 Derived figures
- 4.1 Petrie polygons
- 4.2 Construction
5 See also
6 External links

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 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. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon). The area of a regular octagon of side length a is given by

$A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427125\,a^2.$

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

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

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

The area of a regular octagon can be computed as a truncated square.

The area can also be derived as follows:

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

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:

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

$S=2.414a\,$ (approximately)

The area is then as above:

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

Another simple formula for the area is

$\ A=2ad$

where d is the distance between parallel sides (the same as span S in the diagram).

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

In many parts of the world, stop signs are in the shape of a regular octagon.	Umbrellas often have an octagonal outline.	The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.	The street & block layout of Barcelona's Eixample district is based on non-regular octagons
The famous Vichy Pastilles, octagon-shaped candies.

Derived figures

An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon.	The vertex figure of the uniform polyhedron, great dirhombicosidodecahedron is contained within an irregular 8-sided star polygon, with four edges going through its center.	An octagonal prism contains two octagons.	The truncated square tiling has 2 octagons around every vertex.
The truncated cuboctahedron contains 6 octagons.	An octagonal antiprism contains two octagons.

Petrie polygons

The octagon is the Petrie polygon for eight higher dimensional polytopes, shown in these skew orthogonal projections:

4D				5D
16-cell	Tesseract	Rectified tesseract	24-cell (Rectified 16-cell)	Demipenteract
7D				5D
7-simplex	Rectified 7-simplex	Birectified 7-simplex	Trirectified 7-simplex	Rectified demipenteract

Construction

A regular octagon is constructible using compass and straightedge:

External links

Octagon Calculator
Definition and properties of an octagon With interactive animation
Weisstein, Eric W., "Octagon" from MathWorld.

Polygons

Listed by number of sides

1–10 sides	Henagon (Monogon) · Digon · Triangle (Trigon) · Quadrilateral (Tetragon) · Pentagon · Hexagon · Heptagon · Octagon · Nonagon (Enneagon) · Decagon

11–20 sides	Hendecagon · Dodecagon · Triskaidecagon · Tetradecagon · Pentadecagon · Hexadecagon · Heptadecagon · Octadecagon · Nonadecagon (Enneadecagon) · Icosagon

Others	Apeirogon

Star polygons	Pentagram · Hexagram · Heptagram · Octagram · Enneagram · Decagram · Hendecagram · Dodecagram