Pentagon

A pentagon is a polygon with five sides and five interior angles. In Euclidean geometry, the sum of the interior angles of a pentagon is 540 degrees. In a regular pentagon, all sides and all interior angles are equal, with each interior angle measuring 108 degrees. A regular pentagon has five diagonals, which form a pentagram when drawn inside the shape. The diagonals intersect in such a way that the ratio of a diagonal to a side is the golden ratio, approximately 1.618. The symmetry of a regular pentagon is described by the dihedral group D5, giving fivefold rotational symmetry and five mirror symmetries.

Pentagons appear in various contexts in geometry and art. The fivefold symmetry of the pentagon has influenced styles in decorative patterns and architecture, and pentagonal tilings have been studied in mathematics. A regular pentagon cannot tile the plane by itself, but irregular pentagons can tile the plane under certain conditions.

The term pentagon also refers to the United States Department of Defense headquarters, located in Arlington, Virginia, just outside Washington, D.C. Known as the Pentagon, it is a five-sided building designed to consolidate the offices of the War Department during World War II. Construction began in 1941 and the building opened in 1943. It comprises five floors above ground and two basement levels, with five wings surrounding a central courtyard. It is one of the largest office buildings in the world by floor area and serves as the headquarters for the U.S. Department of Defense and related agencies. The Pentagon was damaged during the September 11, 2001 attacks.