septendecagon

A septendecagon is a polygon with seventeen sides and seventeen angles. 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 whether it is regular or irregular, can be calculated using 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 measures 360 / 17 degrees, approximately 21.18 degrees. The construction of a regular septendecagon was first described by Carl Friedrich Gauss, who proved that it is constructible with a compass and straightedge. This is because 17 is a Fermat prime.

The area of a regular septendecagon with side length 's' can be calculated using the formula A = (17/4) * s^2 * cot(pi/17). The perimeter of a regular septendecagon is simply 17 times the length of one side. Septendecagons are not commonly encountered in everyday geometry but hold mathematical significance due to their constructibility.