Octagon
Regular octagon  

A regular octagon  
Type  Regular polygon 
Edges and vertices  8 
Schläfli symbol  {8}, t{4} 
Coxeter diagram 

Symmetry group  Dihedral (D_{8}), 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 an eightsided polygon or 8gon.
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,
Properties of the general octagon
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}
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° ( radians). The central angle is 45° ( radians).
Area
The area of a regular octagon of side length a is given by
In terms of the circumradius R, the area is
In terms of the apothem r (see also inscribed figure), the area is
These last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be expressed as
where S is the span of the octagon, or the secondshortest 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
The area is then as above:
Expressed in terms of the span, the area is
Another simple formula for the area is
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,
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 may be calculated as
Circumradius and inradius
The circumradius of the regular octagon in terms of the side length a is^{[3]}
and the inradius is
Construction and elementary properties
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.
A regular octagon can be constructed using a straightedge and a compass, as 8 = 2^{3}, a power of two:
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:
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).
Dissection of regular octagon
Coxeter states that every parallelsided 2mgon can be divided into m(m1)/2 rhombs. For the 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.^{[4]}
With 6 rhombs 
Tesseract 
Skew octagon
A skew octagon is a skew polygon with 8 vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zigzag octagon has vertices alternating between two parallel planes.
A regular skew octagon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag 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 higherdimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A_{7}, B_{4}, and D_{5} Coxeter planes.
A_{7}  D_{5}  B_{4}  

7simplex 
5demicube 
16cell 
Tesseract 
Symmetry
The 11 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 Dih_{8} symmetry, order 16. There are 3 dihedral subgroups: Dih_{4}, Dih_{2}, and Dih_{1}, and 4 cyclic subgroups: Z_{8}, Z_{4}, Z_{2}, and Z_{1}, the last implying no symmetry.
r16  

d8 
g8 
p8 
d4 
g4 
p4 
d2 
g2 
p2 
a1 
On the regular octagon, there are 11 distinct symmetries. John Conway labels full symmetry as r16.^{[5]} 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 d8, a isogonal octagon constructed by four mirrors can alternate long and short edges, and p8, 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 seen as directed edges.
Uses of octagons
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, notably the Intelsat Headquarters in Washington D.C., Callam Offices in Canberra, and Octagon Offices in Parramatta, Australia.
Other uses

Umbrellas often have an octagonal outline.

The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.

Janggi uses octagonal pieces.

Japanese lottery machines often have octagonal shape.

Stop sign used in Englishspeaking countries, as well as in most European countries

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 quasioctagonal shape.
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.
Related polytopes
The octagon, as a truncated square, is first in a sequence of truncated hypercubes:
...  
Octagon  Truncated cube  Truncated tesseract  Truncated 5cube  Truncated 6cube  Truncated 7cube  Truncated 8cube  
As an expanded square, it is also first in a sequence of expanded hypercubes:
...  
Octagon  Rhombicuboctahedron  Runcinated tesseract  Stericated 5cube  Pentellated 6cube  Hexicated 7cube  Heptellated 8cube  
See also
 Bumper pool
 Octagon house
 Octagonal number
 Octagram
 Oktogon, a major intersection in Budapest, Hungary
 Rub el Hizb (also known as Al Quds Star and as Octa Star)
 Smoothed octagon
References
 ↑ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
 1 2 Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html
 ↑ Weisstein, Eric. "Octagon." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
External links
 Octagon Calculator
 Definition and properties of an octagon With interactive animation