Hexacontatetragon

Regular hexacontatetragon
A regular hexacontatetragon
Type	Regular polygon
Edges and vertices	64
Schläfli symbol	{64}, t{32}, tt{16}, ttt{8}, tttt{4}
Coxeter diagram
Symmetry group	Dihedral (D64), order 2×64
Internal angle (degrees)	174.375°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a hexacontatetragon (or hexacontakaitetragon) or 64-gon is a sixty-four-sided polygon. (In Greek, the prefix hexaconta- means 60 and tetra- means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees.

The regular hexacontatetragon can be constructed as a truncated triacontadigon, t{32}, a twice-truncated hexadecagon, tt{16}, a thrice-truncated octagon, ttt{8}, a fourfold-truncated square, tttt{4}, and a fivefold-truncated digon, ttttt{2}.

One interior angle in a regular hexacontatetragon is 174 ³⁄₈°, meaning that one exterior angle would be 5 ⁵⁄₈°.

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

and its inradius is

The circumradius of a regular hexacontatetragon is

Since 64 = 2⁶ (a power of two), a regular hexacontatetragon is constructible using a compass and straightedge. As a truncated triacontadigon, it can be constructed by an edge-bisection of a regular triacontadigon.

The regular hexacontatetragon has Dih₆₄dihedral symmetry, order 128, represented by 64 lines of reflection. Dih₆₄ has 6 dihedral subgroups: Dih₃₂, Dih₁₆, Dih₈, Dih₄, Dih₂ and Dih₁ and 7 more cyclic symmetries: Z₆₄, Z₃₂, Z₁₆, Z₈, Z₄, Z₂, and Z₁, with Z_n representing π/n radian rotational symmetry.