Ingeometry, anoctagon (from Ancient Greek ὀκτάγωνον (oktágōnon)'eight angles') is an eight-sidedpolygon or 8-gon.
Aregular octagon hasSchläfli symbol {8}[1] and can also be constructed as a quasiregulartruncatedsquare, t{4}, which alternates two types of edges. A truncated octagon, t{8} is ahexadecagon, {16}. A 3D analog of the octagon can be therhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.
The diagonals of the green quadrilateral are equal in length and at right angles to each other
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 bothequidiagonal andorthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).[2]: Prop. 9
Themidpoint 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
whereS is the span of the octagon, or the second-shortest diagonal; anda 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 are45–45–90 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 sidea, the spanS is
The span, then, is equal to thesilver 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 spanS 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 lengthse 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 regular octagon can be constructed withmeccano 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:
Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(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 theregular 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 aPetrie polygon projection plane of thetesseract. The list (sequenceA006245 in theOEIS) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in theAmmann–Beenker tilings.
A regular skew octagon seen as edges of asquare antiprism, symmetry D4d, [2+,8], (2*4), order 16.
Askew octagon is askew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. Askew zig-zag octagon has vertices alternating between two parallel planes.
Aregular skew octagon isvertex-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 asquare antiprism with the same D4d, [2+,8] symmetry, order 16.
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.
Theregular octagon has Dih8 symmetry, order 16. There are three dihedral subgroups: Dih4, Dih2, and Dih1, and fourcyclic 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 asr16.[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 asg for their central gyration orders. Full symmetry of the regular form isr16 and no symmetry is labeleda1.
The most common high symmetry octagons arep8, anisogonal octagon constructed by four mirrors can alternate long and short edges, andd8, anisotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms areduals 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 theg8 subgroup has no degrees of freedom but can be seen asdirected edges.
Architects such asJohn Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in theIntelsat Headquarters of Washington orCallam Offices in Canberra.
^Alsina, Claudi; Nelsen, Roger B. (2023),A Panoply of Polygons, Dolciani Mathematical Expositions, vol. 58, American Mathematical Society, p. 124,ISBN9781470471842
^Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss, (2008) The Symmetries of Things,ISBN978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
