chiliagon

A chiliagon is a polygon with 1000 sides and 1000 vertices. The term originates from the Greek words "khilioi" meaning thousand and "gonia" meaning angle or corner. Like any regular polygon, a chiliagon can be convex or concave. A regular chiliagon has all sides equal in length and all interior angles equal.

The interior angle of a regular chiliagon can be calculated using the formula (n-2) * 180 / n, where n is the number of sides. For a chiliagon, this means each interior angle measures (1000-2) * 180 / 1000 = 998 * 180 / 1000 = 179.64 degrees.

The sum of the interior angles of any chiliagon, regular or irregular, is (1000-2) * 180 = 179,640 degrees. The exterior angle of a regular chiliagon is 360 / 1000 = 0.36 degrees.

Constructing a regular chiliagon with only a compass and straightedge is impossible. This is because the number of sides, 1000, has a prime factorization of 2^3 * 5^3. According to the Gauss–Wantzel theorem, a regular n-gon can be constructed if and only if n is the product of a power of 2 and distinct Fermat primes. Since 5 is a Fermat prime, but 1000 has 5^3 as a factor, it fails this condition. However, a chiliagon can be constructed using other tools or approximations.