Octagon

For other uses, see Octagon (disambiguation).

Regular octagon
A regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8}
Coxeter diagram
Symmetry group	D8, order 2×8
Internal angle (degrees)	135°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is a regular polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

Regular octagon

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 the circumradius R, the area is

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

In terms of the apothem r (see also inscribed figure), 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 length of a side a, the span S is:

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

$S \approx 2.414a$

The area is then as above:

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

$A \approx 4.828a^2$

Expressed in terms of the span, area is:

$A=2(\sqrt{2}-1)S^2$

$A \approx 0.828S^2$

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).

Construction and elementary properties

A regular octagon may be constructed as follows:

1. Draw a circle and a diameter AOB, where O is the center and A,B are points on the circumference.
2. Draw another diameter COD, perpendicular to AOB.
3. (Note in passing that A,B,C,D are vertices of a square).
4. Draw the bisectors of the right angles AOC and BOC, making two more diameters EOF and GOH.
5. A,B,C,D,E,F,G,H are the vertices of the octagon.

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 8 isosceles triangles, leading to the result: Area = 2*a*a*(sqrt(2) + 1) 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+√2))
• (±(1+√2), ±1).

Uses of octagons

• 

  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

• IMG 1813.jpg

  Vichy Pastilles, octagon-shaped candies.

• 

  Janggi uses octagonal pieces.

• 

  Japanese lottery machines often have octagonal shape.

• 

  Stop sign used in English-speaking countries, as well as in most European countries

Derived figures

• 

  An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon.

• 

  The truncated square tiling has 2 octagons around every vertex.

• 

  An octagonal prism contains two octagonal faces.

• 

  An octagonal antiprism contains two octagonal faces.

• 

  The truncated cuboctahedron contains 6 octagonal faces.

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 demipenteract	Birectified demipenteract	Rectified demipenteract	Demipenteract

See also

• Octagram
• Octagonal number
• Octagon house
• Oktogon, a major intersection in Budapest, Hungary
• Bumper pool
• Rub el Hizb (also known as Al Quds Star and as Octa Star)
• Smoothed octagon

External links

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