Triacontagons

A triacontagon is a polygon with thirty sides and thirty vertices. The term comes from Greek, "triakonta" meaning thirty and "gonia" meaning corner or angle. A regular triacontagon has all sides equal in length and all interior angles equal. The sum of the interior angles of any triacontagon, regular or irregular, is (30-2) * 180 degrees, which equals 4860 degrees.

For a regular triacontagon, each interior angle measures 4860 / 30 = 162 degrees. The exterior angle of a regular triacontagon is 360 / 30 = 12 degrees. The area of a regular triacontagon with side length 's' can be calculated using the formula A = (30/4) * s^2 * cot(pi/30), which simplifies to A = (15/2) * s^2 * cot(6 degrees). Approximating cot(6 degrees) as 9.798, the area is roughly 73.485 * s^2.

A triacontagon can be constructed using a compass and straightedge. Its construction relies on the fact that 30 is a product of distinct Fermat primes (3 and 5) and a power of 2 (2). Specifically, 30 = 2 * 3 * 5. This means that a regular triacontagon is constructible. The internal angles and symmetries of a regular triacontagon are also relevant in various geometric and architectural contexts where complex tessellations or decorative patterns are desired.