Triacontadigon (32-sided polygon)

Key properties

The regular triacontadigon has these straightforward geometric measures and characteristics:

• Interior angle: each interior angle equals (n-2)·180°/n, so for n=32 each interior angle is 168.75°.
• Exterior angle: 360°/32 = 11.25°.
• Diagonals: a 32-gon has n(n-3)/2 diagonals; for n=32 this gives 464 diagonals.
• Symmetry: the regular form has dihedral symmetry D32 of order 64 (32 rotations and 32 reflections).
• Coordinates and area: vertices on a circumcircle are located at (cos(2πk/32), sin(2πk/32)) for k=0..31. The area of a regular n-gon can be written A = (n/4)s^2 cot(π/n) in terms of side length s.

Constructibility and star forms

Because 32 is a power of two, a regular triacontadigon is constructible with compass and straightedge: successive angle bisections from constructible polygons yield it. Regular star polygons based on 32 vertices also exist; regular star figures take the form {32/k} and produce nontrivial single-cycle stars when k is odd and between 3 and 15 (for example {32/3}, {32/5}, {32/7}, etc.).

History, naming, and uses

Polygon names derive from classical Greek numerical prefixes combined with -gon. In practice the simple label "32-gon" is common in mathematics and engineering. Regular 32-gons appear in design and computation where a circle is approximated by a polygon (for instance in computer graphics, machining and tiling patterns). Their high number of sides gives a close visual approximation of a circle while remaining a simple straight-edged figure.

Notable facts and distinctions

Compared with polygons with fewer sides, the triacontadigon reduces angular steps (exterior angle 11.25°) and increases the number of diagonals dramatically (464). It sits among polygons that are straightforward to construct and to subdivide into isosceles triangles from the center, making it useful pedagogically when illustrating symmetry and polygonal area formulas. For further general background on polygons see 32-sided polygon resources.