Septendecagons

A septendecagon is a polygon with 17 sides and 17 vertices. In a regular septendecagon, all sides are of equal length and all interior angles are equal. The sum of the interior angles of any septendecagon, regardless of regularity, is given by the formula (n-2) * 180 degrees, where n is the number of sides. For a septendecagon, this sum is (17-2) * 180 = 15 * 180 = 2700 degrees.

In a regular septendecagon, each interior angle measures 2700 / 17 degrees, which is approximately 158.82 degrees. The exterior angle of a regular septendecagon is 360 / 17 degrees, approximately 21.18 degrees. The construction of a regular septendecagon using only a compass and straightedge was a significant achievement in geometry. It was proven possible by Carl Friedrich Gauss in 1796. This discovery was a major step towards understanding which regular polygons are constructible.

The area of a regular septendecagon with side length 'a' can be calculated using the formula A = (17/4) * a^2 * cot(pi/17). The perimeter of a regular septendecagon is simply 17 times its side length. Septendecagons, while mathematically significant due to their constructibility, are not commonly encountered in everyday objects or natural formations.