Regular octagon | |
---|---|

Type | Regular polygon |

Edges and vertices | 8 |

Schläfli symbol | {8}, t{4} |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral (D_{8}), 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.

- Properties
- Regularity
- Area
- Circumradius and inradius
- Diagonality
- Construction
- Standard coordinates
- Dissectibility
- Skew
- Petrie polygons
- Symmetry
- Use
- Derived figures
- Related polytopes
- See also
- References
- External links

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.

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 }

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).

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 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

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

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

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

and the inradius is

(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

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: ;
- Medium diagonal: ; (
*silver ratio*times a) - Long diagonal: .

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:

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:

for an octagon of side *a*.

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).

8-cube projection | 24 rhomb dissection | |
---|---|---|

Regular | Isotoxal | |

Coxeter states that every zonogon (a 2*m*-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.

Tesseract | 4 rhombs and 2 square |

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.

The regular skew octagon is the Petrie polygon for these higher-dimensional 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} | |
---|---|---|---|

7-simplex | 5-demicube | 16-cell | Tesseract |

The *regular octagon* has Dih_{8} symmetry, order 16. There are three dihedral subgroups: Dih_{4}, Dih_{2}, and Dih_{1}, and four 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 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.

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.
- Janggi uses octagonal pieces.
- Japanese lottery machines often have octagonal shape.
- 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 restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.
- Chair from A la Ronde, with octagonal seats and backs (set of eight)

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

... | |||||||

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

- Bumper pool
- Hansen's small octagon
- Octagon house
- Octagonal number
- Octagram
- Octahedron, 3D shape with eight faces.
- Oktogon, a major intersection in Budapest, Hungary
- Rub el Hizb (also known as Al Quds Star and as Octa Star)
- Smoothed octagon

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 **C _{5}**,

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.

- ↑ 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, 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)

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