pentagons

A pentagon is a polygon with five sides. In a simple pentagon, the sum of the interior angles equals 540 degrees. If all interior angles are less than 180 degrees, the pentagon is convex; if one or more angles exceed 180 degrees, the pentagon is concave.

Regular pentagon: all sides and all interior angles are equal; each interior angle is 108 degrees and each exterior angle is 72 degrees. The ratio of a diagonal to a side in a regular pentagon equals the golden ratio φ ≈ 1.618. The five diagonals form a pentagram, a star polygon. The symmetry group of a regular pentagon is the dihedral group D5, of order 10.

Area and measurements: for a regular pentagon with side length s, the area is A = (5 s^2)/(4 tan(π/5)) = (1/4) sqrt(5(5+2√5)) s^2. If the apothem a is known, the area is A = (1/2) a P, where P is the perimeter; for a regular pentagon, a = s/(2 tan(π/5)).

Pentagons in tiling and design: while a single pentagon cannot tile the plane by itself, certain convex pentagons can tile the plane in specific arrangements. Pentagonal shapes also appear in nature and architecture, influencing patterns in chemistry, crystallography, and decorative geometry.