pentadecagon

A pentadecagon is a polygon with fifteen sides and fifteen vertices. In a regular pentadecagon, all sides are equal in length and all interior angles are equal. The sum of the interior angles of any pentadecagon, regardless of whether it is regular or irregular, is 2340 degrees. Each interior angle of a regular pentadecagon measures 156 degrees. The exterior angle of a regular pentadecagon is 24 degrees.

A regular pentadecagon can be constructed using a straightedge and compass, as 15 is a product of distinct Fermat primes (3 and 5). The area of a regular pentadecagon with side length 'a' can be calculated using the formula: Area = (15/4) * a^2 * cot(pi/15). The radius of the circumscribed circle for a regular pentadecagon is R = a / (2 * sin(pi/15)), and the radius of the inscribed circle is r = a / (2 * tan(pi/15)). The term "pentadecagon" is derived from the Greek words "pente" meaning five and "deca" meaning ten, combined with "gonia" meaning angle.