In geometry, an octagon (from the Greek ὀκτάγωνον
oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.
A regular octagon has
Contents 1 Properties of the general octagon 2 Regular octagon 2.1 Area
2.2
3 Skew octagon 3.1 Petrie polygons 4 Symmetry 5 Uses of octagons 5.1 Other uses 6 Derived figures 6.1 Related polytopes 7 See also 8 References 9 External links Properties of the general octagon[edit] 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 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[edit]
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
3 π 4 displaystyle scriptstyle frac 3pi 4 radians). The central angle is 45° ( π 4 displaystyle scriptstyle frac pi 4 radians). Area[edit] The area of a regular octagon of side length a is given by A = 2 cot π 8 a 2 = 2 ( 1 + 2 ) a 2 ≃ 4.828 a 2 . displaystyle A=2cot frac pi 8 a^ 2 =2(1+ sqrt 2 )a^ 2 simeq 4.828,a^ 2 . In terms of the circumradius R, the area is A = 4 sin π 4 R 2 = 2 2 R 2 ≃ 2.828 R 2 . displaystyle A=4sin frac pi 4 R^ 2 =2 sqrt 2 R^ 2 simeq 2.828,R^ 2 . In terms of the apothem r (see also inscribed figure), the area is A = 8 tan π 8 r 2 = 8 ( 2 − 1 ) r 2 ≃ 3.314 r 2 . displaystyle A=8tan frac pi 8 r^ 2 =8( sqrt 2 -1)r^ 2 simeq 3.314,r^ 2 . These last two coefficients bracket the value of pi, the area of the unit circle. The area of a regular octagon can be computed as a truncated square. The area can also be expressed as A = S 2 − a 2 , 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 S = a 2 + a + a 2 = ( 1 + 2 ) a ≈ 2.414 a . displaystyle S= frac a sqrt 2 +a+ frac a sqrt 2 =(1+ sqrt 2 )aapprox 2.414a. The area is then as above: A = ( ( 1 + 2 ) a ) 2 − a 2 = 2 ( 1 + 2 ) a 2 ≈ 4.828 a 2 . 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 A = 2 ( 2 − 1 ) S 2 ≈ 0.828 S 2 . displaystyle A=2( sqrt 2 -1)S^ 2 approx 0.828S^ 2 . Another simple formula for the area is A = 2 a S . 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, a ≈ S / 2.414. displaystyle aapprox 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 / 2 , displaystyle e=a/ sqrt 2 , may be calculated as e = ( S − a ) / 2. displaystyle ,!e=(S-a)/2.
R = ( 4 + 2 2 2 ) a , displaystyle R=left( frac sqrt 4+2 sqrt 2 2 right)a, and the inradius is r = ( 1 + 2 2 ) a . displaystyle r=left( frac 1+ sqrt 2 2 right)a. Diagonals[edit] 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:[citation needed] Short diagonal: a 2 + 2 displaystyle a sqrt 2+ sqrt 2 ; Medium diagonal: ( 1 + 2 ) a displaystyle (1+ sqrt 2 )a ; Long diagonal: a 4 + 2 2 displaystyle a sqrt 4+2 sqrt 2 . Construction and elementary properties[edit] 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.
A regular octagon can be constructed using a straightedge and a compass, as 8 = 23, 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: Area = 2 a 2 ( 2 + 1 ) displaystyle text
for an octagon of side a. Standard coordinates[edit] 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[edit]
Regular Isotoxal
Regular octagon dissected Tesseract 4 rhombs and 2 square Skew octagon[edit] A regular skew octagon seen as edges of a square antiprism, symmetry D4d, [2+,8], (2*4), order 16. 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 zig-zag octagon has vertices alternating
between two parallel planes.
A regular skew octagon is vertex-transitive with equal edge lengths.
In 3-dimensions it will be 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[edit]
The regular skew octagon is the
A7 D5 B4 7-simplex 5-demicube 16-cell Tesseract Symmetry[edit] 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 Dih8 symmetry, order 16. There are 3 dihedral subgroups: Dih4, Dih2, and Dih1, and 4 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 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 p8, a 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 seen as directed edges. Uses of octagons[edit] The octagonal floor plan, Dome of the Rock. The octagonal shape is used as a design element in architecture. The
Umbrellas often have an octagonal outline. The famous
The street & block layout of Barcelona's
Japanese lottery machines often have octagonal shape.
An icon of a stop sign with a hand at the middle. The trigrams of the Taoist bagua are often arranged octagonally Famous octagonal gold cup from the Belitung shipwreck Classes at
The
Derived figures[edit] 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[edit] The octagon, as a truncated square, is first in a sequence of truncated hypercubes: Truncated hypercubes ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube 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 See also[edit] Bumper pool
References[edit] ^ 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 ^ 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[edit]
v t e Polygons Regular List 1–10 sides Monogon Digon Triangle Equilateral Isosceles Quadrilateral Square Rectangle Rhombus Parallelogram Trapezoid Kite Pentagon
Hexagon
Heptagon
Octagon
11–20 sides Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon 21–100 sides (selected)
>100 sides 120-gon
257-gon
360-gon
Star polygons (5–12 sides) Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hend |