20gon

A 20gon is a polygon with 20 sides and 20 vertices. In Euclidean geometry, a 20gon can be either regular or irregular. A regular 20gon has all sides equal in length and all interior angles equal in measure. The sum of the interior angles of any 20gon is given by the formula (n-2) * 180 degrees, where n is the number of sides. For a 20gon, this sum is (20-2) * 180 = 18 * 180 = 3240 degrees. In a regular 20gon, each interior angle measures 3240 / 20 = 162 degrees. The exterior angle of a regular 20gon is 360 / 20 = 18 degrees.

The construction of a regular 20gon using only a compass and straightedge is possible, as 20 can be expressed as 2^2 * 5, and 5 is a Fermat prime. This means that the regular 20gon is a constructible polygon. The perimeter of a regular 20gon inscribed in a circle of radius r is approximately 20 * 2r * sin(180/20) = 40r * sin(9 degrees). The area of a regular 20gon inscribed in a circle of radius r is given by (1/2) * n * r^2 * sin(360/n), which for a 20gon is 10r^2 * sin(18 degrees).

The term "20gon" is often used in contexts where the specific number of sides is important, such as in geometry problems, tessellations, or discussions of polyhedra and their faces. While not as commonly encountered as polygons with fewer sides, the 20gon has specific mathematical properties derived from its number of sides.