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Regular octadecagon
Regular polygon 18 annotated.svg
A regular octadecagon
Type Regular polygon
Edges and vertices 18
Schläfli symbol {18}
Coxeter diagram CDel node 1.pngCDel 18.pngCDel node.png
CDel node 1.pngCDel 9.pngCDel node 1.png
Symmetry group Dihedral (D18), order 2×18
Internal angle (degrees) 160°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

An octadecagon (or octakaidecagon[1]) is a polygon with 18 sides and 18 vertices.[2]

A regular octadecagon has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges.


A regular octadecagon cannot be constructed using a compass and straightedge.[3] However, it is constructible using neusis, or an angle trisector.


3.9.18 vertex.png
A regular triangle, enneagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of regular polygons with this property.[4] However, this pattern cannot be extended to an Archimedean tiling of the plane: because the triangle and the enneagon both have an odd number of sides, neither of them can be completely surrounded by a ring alternating the other two kinds of polygon.

Related figures[edit]

An octadecagram is an 18-sided star polygon, represented by symbol {18/n}. There are two regular star polygons: {18/5} and {18/7}, using the same points, but connecting every fifth or seventh points. There are also five compounds: {18/2} is reduced to 2{9} or two enneagons, {18/3} is reduced to 3{6} or three hexagons, {18/4} and {18/8} are reduced to 2{9/2} and 2{9/4} or two enneagrams, {18/6} is reduced to 6{3} or 6 equilateral triangles, and finally {18/9} is reduced to 9{2} as nine digons.

n 1 2 3 4 5 6 7 8 9
Form Convex polygon Compounds Star polygon Compound Star polygon Compound
Image Regular polygon 18.svg
= {18}
Regular star figure 2(9,1).svg
= 2{9}
Regular star figure 3(6,1).svg
= 3{6}
Regular star figure 2(9,2).svg
= 2{9/2}
Regular star polygon 18-5.svg
Regular star figure 6(3,1).svg
= 6{3}
Regular star polygon 18-7.svg
Regular star figure 2(9,4).svg
= 2{9/4}
Regular star figure 9(2,1).svg
= 9{2}
Interior angle 160° 140° 120° 100° 80° 60° 40° 20°

Deeper truncations of the regular enneagon and enneagrams can produce isogonal (vertex-transitive) intermediate octadecagram forms with equally spaced vertices and two edge lengths. Other truncations form double coverings: t{9/8}={18/8}=2{9/4}, t{9/4}={18/4}=2{9/2}, t{9/2}={18/2}=2{9}.[5]

Vertex-transitive truncations of enneagon and enneagrams
Quasiregular isogonal Quasiregular
Double covering
Regular polygon truncation 9 1.svg
Regular polygon truncation 9 2.svg Regular polygon truncation 9 3.svg Regular polygon truncation 9 4.svg Regular polygon truncation 9 5.svg Regular star polygon 9-4.svg
Regular star truncation 9-5 1.svg
Regular star truncation 9-5 2.svg Regular star truncation 9-5 3.svg Regular star truncation 9-5 4.svg Regular star truncation 9-5 5.svg Regular star polygon 9-2.svg
Regular star truncation 9-7 1.svg
Regular star truncation 9-7 2.svg Regular star truncation 9-7 3.svg Regular star truncation 9-7 4.svg Regular star truncation 9-7 5.svg Regular polygon 9.svg

Petrie polygons[edit]

The regular octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these skew orthogonal projections from Coxeter planes:

A17 B9 D10 E7
17-simplex t0.svg
9-cube t8.svg
9-cube t0.svg
10-cube t9 B9.svg
Up2 3 21 t0 E7.svg
Up2 2 31 t0 E7.svg
Up2 1 32 t0 E7.svg


  1. ^ Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 86, ISBN 9781930190092 .
  2. ^ Adams, Henry (1907), Cassell's Engineer's Handbook: Comprising Facts and Formulæ, Principles and Practice, in All Branches of Engineering, D. McKay, p. 528 .
  3. ^ Conway, John B. (2010), Mathematical Connections: A Capstone Course, American Mathematical Society, p. 31, ISBN 9780821849798 .
  4. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134 .
  5. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links[edit]