# Octagon

(Redirected from Octogon)
For other uses, see Octagon (disambiguation).
"Octagonal" redirects here. For other uses, see Octagonal (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 polygon that has eight sides.

## 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. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles of any octagon is 1080° (as with all polygons, the external angles total 360°). 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

$\ A=2aS.$

More often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above:

$a \approx S/2.414$

The two end lengths e on each side, as well as being e=a/sqrt{2}, may be calculated as:

$\,\!e=(S-a)/2$

## 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:

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

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

The octagonal floor plan, Dome of the Rock.

The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna), Castel del Monte (Apulia), Florence Baptistery, Zum Friedefürsten church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral.

## Derived figures

### Related polytopes

The octagon, as a truncated square, is first in a sequence of truncated hypercubes:

 ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube

### 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.