23gon

A 23gon is a polygon with 23 sides and 23 vertices. In Euclidean geometry, the sum of the interior angles of a 23gon is (23-2) * 180 degrees, which equals 21 * 180 degrees or 3780 degrees. Each interior angle of a regular 23gon measures 3780 / 23 degrees, approximately 164.35 degrees. Similarly, the sum of the exterior angles of any convex 23gon is always 360 degrees. Each exterior angle of a regular 23gon measures 360 / 23 degrees, approximately 15.65 degrees.

A regular 23gon cannot be constructed using only a compass and straightedge. This is because the constructibility of a regular n-gon depends on the prime factorization of n. A regular n-gon is constructible if and only if n is the product of a power of 2 and distinct Fermat primes. The number 23 is a prime number, but it is not a Fermat prime (Fermat primes are of the form 2^(2^k) + 1, such as 3, 5, 17, 257, and 65537). Therefore, a regular 23gon is not constructible with compass and straightedge.

The number of diagonals in a 23gon is given by the formula n(n-3)/2, where n is the number of sides. For a 23gon, this is 23(23-3)/2 = 23 * 20 / 2 = 230 diagonals. The 23gon is a decagon related polygon, meaning it is a polygon with a large number of sides, often considered in theoretical geometric contexts or for computational purposes.