Icosikaitetragon (Icositetragon)

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).

Further reading

For a simple diagram and basic definitions, see a general polygon reference: 24-sided polygon. More advanced discussions treat constructibility, star polygon families, and algebraic expressions for trigonometric values needed to derive exact side and area formulas.