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
* 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 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). :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. :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
AREA The area of a regular octagon of side length _a_ is given by A = 2 cot 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 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 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.} CIRCUMRADIUS AND INRADIUS The circumradius of the regular octagon in terms of the side length _a_ is 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.} CONSTRUCTION AND ELEMENTARY PROPERTIES 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:
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
Regular octagon dissected With 6 rhombs
SKEW OCTAGON A regular skew octagon seen as edges of a square antiprism , symmetry D4d, , (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, symmetry, order 16. PETRIE POLYGONS The regular skew octagon is the
A7 D5 B4
SYMMETRY 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. 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 The octagonal floor plan, Dome of the Rock. The octagonal shape is used as a design element in architecture. The
Architects such as John Andrews have used octagonal floor layouts in
buildings for functionally separating office areas from building
services, notably the
OTHER USES * Umbrellas often have an octagonal outline. * The famous
The street ">
Japanese lottery machines often have octagonal shape. *
The trigrams of the Taoist _bagua _ are often arranged octagonally * Famous octagonal gold cup from the
Classes at
The
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. * The omnitruncated cubic honeycomb RELATED POLYTOPES The _octagon_, as a truncated square , is first in a sequence of truncated hypercubes : Truncated hypercubes ... Octagon
As an expanded square, it is also first in a sequence of expanded hypercubes: Expanded hypercubes ... Octagon
SEE ALSO *
REFERENCES * ^ 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
* ^
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