Hexadecagon

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Regular hexadecagon
A regular hexadecagon
Type	Regular polygon
Edges and vertices	16
Schläfli symbol	{16}
Coxeter diagram
Symmetry group	D₁₆, order 2×16
Internal angle (degrees)	157.5°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In mathematics, a hexadecagon (sometimes called a hexakaidecagon) is a polygon with 16 sides and 16 vertices.^[1]

Regular hexadecagon

A regular hexadecagon is constructible with a compass and straightedge.

Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees.

Construction

A regular hexadecagon is constructible using compass and straightedge:

Construction of a regular hexadecagon

Area

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

$A=4t^{2}\cot {\frac {\pi }{16}}=4t^{2}({\sqrt {2}}+1)({\sqrt {4-2{\sqrt {2}}}}+1)$

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:

$A=r^{2}\cdot {\frac {2}{1}}\cdot {\frac {2}{{\sqrt {2}}}}\cdot {\frac {2}{{\sqrt {2+{\sqrt {2}}}}}}=4r^{2}{\sqrt {2-{\sqrt {2}}}}.$

Petrie polygons

The regular hexadecagon is the Petrie polygon for many higher dimensional polytopes, shown in these skew orthogonal projections, including:

A₁₅	15-simplex
B₈	8-orthoplex	Rectified 8-orthoplex	Birectified 8-orthoplex	Trirectified 8-orthoplex	Trirectified 8-cube	Birectified 8-cube	Rectified 8-cube	8-cube
D₉	t₇(1₆₁)	t₆(1₆₁)	t₅(1₆₁)	t₄(1₆₁)	t₃(1₆₁)	t₂(1₆₁)	t₁(1₆₁)	9-demicube (1₆₁)

References

↑ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1365. ISBN 9781420035223.

External links

Weisstein, Eric W., "Hexadecagon", MathWorld.

v t e Regular polygons

Listed by number of sides

1–10 sides	Henagon (Monogon) Digon Equilateral triangle Square Pentagon Hexagon Heptagon Octagon Nonagon (Enneagon) Decagon

11–20 sides	Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Nonadecagon (Enneadecagon) Icosagon

Others	Triacontagon Chiliagon Megagon Apeirogon

Star polygons	Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hendecagram Dodecagram

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