Octagon

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Regular octagon
Regular polygon 8.svg
A regular octagon
Type Regular polygon
Edges and vertices 8
Schläfli symbol {8}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.png
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}.

Contents

Regular octagon [edit]

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 [edit]

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 [edit]

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 [edit]

Derived figures [edit]

Petrie polygons [edit]

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
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
16-cell
4-cube t2.svg
Rectified 16-cell
4-cube t1.svg
Rectified tesseract
4-cube t0.svg
Tesseract
D5 5-demicube t3 D5.svg
Trirectified demipenteract
5-demicube t2 D5.svg
Birectified demipenteract
5-demicube t1 D5.svg
Rectified demipenteract
5-demicube t0 D5.svg
Demipenteract

See also [edit]

References [edit]

External links [edit]