Icosikaitetragon (Icositetragon)

A 24-sided polygon (icosikaitetragon or icositetragon). Definition, regular-case properties, construction, symmetry, formulas and notable examples.

Overview

An icosikaitetragon, also called an icositetragon, is a polygon with 24 sides and 24 vertices. The name derives from Greek roots: "icosi-" (twenty) plus "tetra-" (four), often written as icosi+kai+tetragon in older sources. A convex 24-sided polygon has interior angles that sum to (24-2)×180 = 3960 degrees; each interior angle of a regular 24-gon measures 165° and each central angle measures 15°.

Key properties

Number of sides/vertices: 24.
Sum of interior angles: 3960°; each interior angle (regular): 165°.
Exterior/central angle (regular): 15°.
Number of diagonals: n(n−3)/2 = 252.
Schläfli symbol (regular): {24}.
Symmetry group (regular): dihedral group D24 of order 48 (24 rotations and 24 reflections).

Formulas and coordinates

A regular 24-gon of side length s has perimeter P = 24s and area A = (24 s^2)/(4 tan(π/24)) = (6 s^2)/tan(π/24). If the polygon is inscribed in a circle of radius R, its area can be written A = (24/2) R^2 sin(2π/24) = 12 R^2 sin(15°). For unit circumradius R = 1 this simplifies to A = 12 sin 15° = 3(√6 − √2), since sin 15° = (√6 − √2)/4. Vertex coordinates for a centered regular 24-gon lie at the 24th roots of unity: e^{2πik/24} for k = 0,…,23.

Construction and regular star forms

The regular 24-gon is constructible with straightedge and compass because 24 = 3·2^3 and 3 is a Fermat prime; equivalently one may construct a 15° central angle by standard constructions (e.g., combine a 60° and a 45° or bisect angles). Beyond the convex form {24}, several regular star polygons can be drawn using the same 24 evenly spaced vertices. Notable regular star forms include {24/5}, {24/7} and {24/11}, which connect every 5th, 7th or 11th vertex in turn to produce star figures.

Uses, examples and distinctions

Large regular polygons such as the 24-gon are often used to approximate circles in engineering, computer graphics and geometric design. They also appear in tiling and decorative motifs where a many-sided form is desired. Do not confuse the planar icosikaitetragon with higher-dimensional objects whose names include 24 (for example, the 24-cell is a four-dimensional regular polytope and is unrelated except by the shared number).