# Octagon

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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 the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.

## Contents

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

### Area

The area of a regular octagon of side length a is given by

${\displaystyle A=2\cot {\frac {\pi }{8}}a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828\,a^{2}.}$

In terms of the circumradius R, the area is

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

In terms of the apothem r (see also inscribed figure), the area is

${\displaystyle A=8\tan {\frac {\pi }{8}}r^{2}=8({\sqrt {2}}-1)r^{2}\approx 3.314\,r^{2}.}$

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

The area can also be expressed as

${\displaystyle \,\!A=S^{2}-a^{2},}$

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

${\displaystyle S={\frac {a}{\sqrt {2}}}+a+{\frac {a}{\sqrt {2}}}=(1+{\sqrt {2}})a\approx 2.414a.}$

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

The area is then as above:

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

Expressed in terms of the span, the area is

${\displaystyle A=2({\sqrt {2}}-1)S^{2}\approx 0.828S^{2}.}$

Another simple formula for the area is

${\displaystyle \ A=2aS.}$

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,

${\displaystyle a\approx S/2.414.}$

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 ${\displaystyle e=a/{\sqrt {2}},}$ may be calculated as

${\displaystyle \,\!e=(S-a)/2.}$

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

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

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

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

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

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

### Construction

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

1. Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
2. Draw another diameter GOC, perpendicular to AOE.
3. (Note in passing that A,C,E,G are vertices of a square).
4. Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
5. 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:

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

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 projection24 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 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.

 Tesseract 4 rhombs and 2 square

## 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 D4d, [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.

A7D5B4

7-simplex

5-demicube

16-cell

Tesseract

## 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.

## Derived figures

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

 Image Name Coxeter diagram Vertex figure ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube ( )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:

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

## Related Research Articles

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.

In geometry, the deltoidal icositetrahedron is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

In mathematics, a hexadecagon is a sixteen-sided polygon.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

## References

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