Hexacontagon

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Regular hexacontagon
Regular polygon 60.svg
A regular hexacontagon
TypeRegular polygon
Edges and vertices60
Schläfli symbol{60}, t{30}, tt{15}
Coxeter diagramCDel node 1.pngCDel 6.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 3x.pngCDel 0x.pngCDel node 1.png
Symmetry groupDihedral (D60), order 2×60
Internal angle (degrees)174°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

A hexacontagon or 60-gon is a shape with 60 sides and 60 corners.[1][2]

Regular hexacontagon[change | change source]

A regular hexacontagon is represented by Schläfli symbol {60} and also can be constructed as a truncated triacontagon, t{30}, or a twice-truncated pentadecagon, tt{15}. A truncated hexacontagon, t{60}, is a 120-gon, {120}.

One interior angle in a regular hexacontagon is 174°, meaning that one exterior angle would be 6°.

Area[change | change source]

The area of a regular hexacontagon is (with t = edge length)

and its inradius is

The circumradius of a regular hexacontagon is

This means that the trigonometric functions of π/60 can be expressed in radicals.

Constructible[change | change source]

Since 60 = 22 × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge.[3] As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

Dissection[change | change source]

60-gon with 1740 rhombs

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexacontagon, m=30, and it can be divided into 435: 15 squares and 14 sets of 30 rhombs. This decomposition is based on a Petrie polygon projection of a 30-cube.

Examples
60-gon rhombic dissection.svg
60-gon rhombic dissection2.svg
60-gon rhombic dissectionx.svg


References[change | change source]

  1. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 78, ISBN 9781438109572.
  2. The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. Constructible Polygon
  4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141