Hexadecagon facts for kids
Regular hexadecagon  

A regular hexadecagon


Type  Regular polygon 
Edges and vertices  16 
Schläfli symbol  {16}, t{8}, tt{4} 
Coxeter diagram  
Symmetry group  Dihedral (D_{16}), order 2×16 
Internal angle (degrees)  157.5° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
An hexadecagon or hekkaidecagon is a shape with 16 sides and 16 corners.
Regular hexadecagon
A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twicetruncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.
Area
The area of a regular hexadecagon with edge length t is
Because the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius R by truncating Viète's formula:
Since the area of the circumcircle is the regular hexadecagon fills approximately 97.45% of its circumcircle.
Dissection
Coxeter states that every parallelsided 2mgon can be divided into m(m1)/2 rhombs. For the regular hexadecagon, m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petrie polygon projection of a 8cube, with 28 of 1792 faces. The list A006245 enumerates the number of solutions as 1232944, including up to 16fold rotations and chiral forms in reflection.
Skew hexadecagon
A skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A skew zigzag hexadecagon has vertices alternating between two parallel planes.
A regular skew hexadecagon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D_{8d}, [2^{+},16] symmetry, order 32. The octagrammic antiprism, s{2,16/3} and octagrammic crossedantiprism, s{2,16/5} also have regular skew octagons.
In art
In the early 16th century, Raphael was the first to construct a perspective image of a regular hexadecagon: the tower in his painting The Marriage of the Virgin has 16 sides, elaborating on an eightsided tower in a previous painting by Pietro Perugino.
Hexadecagrams (16sided star polygons) are included in the Girih patterns in the Alhambra.
Irregular hexadecagons
An octagonal star can be seen as a concave hexadecagon:
