Octagon

Regular octagon
Regular octagon
	A regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8}, t{4}
Coxeter–Dynkin diagrams	;
Symmetry group	Dihedral (D8), order 2×8
Internal angle (degrees)	135°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, an octagon (from Ancient Greek ὀκτάγωνον (oktágōnon) 'eight angles') is an eight-sided polygon or 8-gon.

A regular octagon has Schläfli symbol {8} ^[1] and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.

Properties

The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.

If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).^[2]^{: Prop. 9}

The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.^[2]^{: Prop. 10}

Regularity

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° ( $\scriptstyle {\frac {3\pi }{4}}$ radians). The central angle is 45° ( $\scriptstyle {\frac {\pi }{4}}$ radians).

Area

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}\approx 4.828\,a^{2}.

In terms of the circumradius R, the area is

A=4\sin {\frac {\pi }{4}}R^{2}=2{\sqrt {2}}R^{2}\approx 2.828\,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}\approx 3.314\,r^{2}.

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

The area can also be expressed as

\,\!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 overlap with the four sides of the square) and then takes the corner triangles (these are 45–45–90 triangles) and places 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\approx 2.414a.

The span, then, is equal to the silver ratio times the side, a.

The area is then as above:

A=((1+{\sqrt {2}})a)^{2}-a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828a^{2}.

Expressed in terms of the span, the area is

A=2({\sqrt {2}}-1)S^{2}\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 (the leg lengths of the triangles (green in the image) truncated from the square), as well as being $e=a/{\sqrt {2}},$ may be calculated as

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

Circumradius and inradius

The circumradius of the regular octagon in terms of the side length a is^[3]

R=\left({\frac {\sqrt {4+2{\sqrt {2}}}}{2}}\right)a\approx 1.307a,

and the inradius is

r=\left({\frac {1+{\sqrt {2}}}{2}}\right)a\approx 1.207a.

(that is one-half the silver ratio times the side, a, or one-half the span, S)

The inradius can be calculated from the circumradius as

r=R\cos {\frac {\pi }{8}}

Diagonality

The regular octagon, in terms of the side length a, has three different types of diagonals:

Short diagonal;
Medium diagonal (also called span or height), which is twice the length of the inradius;
Long diagonal, which is twice the length of the circumradius.

The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:^[4]

Short diagonal: $a{\sqrt {2+{\sqrt {2}}}}$ ;
Medium diagonal: $(1+{\sqrt {2}})a$ ; (silver ratio times a)
Long diagonal: $a{\sqrt {4+2{\sqrt {2}}}}$ .

Construction

building a regular octagon by folding a sheet of paper

A regular octagon at a given circumcircle may be constructed as follows:

Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
Draw another diameter GOC, perpendicular to AOE.
(Note in passing that A,C,E,G are vertices of a square).
Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
A,B,C,D,E,F,G,H are the vertices of the octagon.

Octagon at a given circumcircle

Octagon at a given side length, animation
(The construction is very similar to that of hexadecagon at a given side length.)

A regular octagon can be constructed using a straightedge and a compass, as 8 = 2³, a power of two:

The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required.

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 eight isosceles triangles, leading to the result:

{\text{Area}}=2a^{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).

Dissectibility

8-cube projection	24 rhomb dissection
	Regular	Isotoxal

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.^[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list (sequence A006245 in the OEIS) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings.

Regular octagon dissected
Tesseract	4 rhombs and 2 squares

Skew

A skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes.

A regular skew octagon is vertex-transitive with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D_4d, [2⁺,8] symmetry, order 16.

Petrie polygons

The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

A₇	D₅	B₄
7-simplex	5-demicube	16-cell	Tesseract

Symmetry

Symmetry
	The eleven symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.

The regular octagon has Dih₈ symmetry, order 16. There are three dihedral subgroups: Dih₄, Dih₂, and Dih₁, and four cyclic subgroups: Z₈, Z₄, Z₂, and Z₁, the last implying no symmetry.

Example octagons by symmetry
r16
d8	g8	p8
d4	g4	p4
d2	g2	p2
a1

On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as r16.^[6] The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r16 and no symmetry is labeled a1.

The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8, an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g8 subgroup has no degrees of freedom but can be seen as directed edges.

Use

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, Italia), Castel del Monte (Apulia, Italia), 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.

Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in the Intelsat Headquarters of Washington or Callam Offices in Canberra.

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
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
The trigrams of the Taoist bagua are often arranged octagonally
Famous octagonal gold cup from the Belitung shipwreck
Classes at Shimer College are traditionally held around octagonal tables
The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.
The movement of the analog stick(s) of the Nintendo 64 controller, the GameCube controller, the Wii Nunchuk and the Classic Controller is surrounded by a rotated octagonal area, helping the user aim the stick in cardinal directions while still allowing circular freedom.
Chair from A la Ronde, with octagonal seats and backs (set of eight)

Derived figures

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.
The omnitruncated cubic honeycomb

Related polytopes

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

Truncated hypercubes
Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

As an expanded square, it is also first in a sequence of expanded hypercubes:

Expanded hypercubes
							...
Octagon	Rhombicuboctahedron	Runcinated tesseract	Stericated 5-cube	Pentellated 6-cube	Hexicated 7-cube	Heptellated 8-cube

References

^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
^ ^a ^b Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html
^ Weisstein, Eric. "Octagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html
^ Alsina, Claudi; Nelsen, Roger B. (2023), A Panoply of Polygons, Dolciani Mathematical Expositions, vol. 58, American Mathematical Society, p. 124, ISBN 9781470471842
^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)

External links

Octagon Calculator
Definition and properties of an octagon With interactive animation

[1] Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.

[Oai-2] Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html

[3] Weisstein, Eric. "Octagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html

[4] Alsina, Claudi; Nelsen, Roger B. (2023), A Panoply of Polygons, Dolciani Mathematical Expositions, vol. 58, American Mathematical Society, p. 124, ISBN 9781470471842

[5] Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

[6] John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)

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