Pentagon (geometry)
Definition and Basic Properties
A pentagon is a polygon with five sides and five vertices. The term "pentagon" comes from the Greek words "pente" meaning five and "gonia" meaning angle. The sum of the interior angles of a pentagon is always 540 degrees, regardless of whether it is regular or irregular. A regular pentagon, where all sides and angles are equal, has each interior angle measuring 108 degrees.
Regular Pentagon
A regular pentagon is a pentagon where all sides are of equal length, and all interior angles are equal, each measuring 108 degrees. The regular pentagon is one of the most commonly encountered types of pentagons in both practical applications and theoretical mathematics. The regular pentagon is also a type of convex polygon, meaning that a line segment drawn between any two points in the polygon will always lie entirely within the polygon.
Construction
The construction of a regular pentagon can be achieved using a compass and straightedge, following a process that has been known since ancient times. The process involves creating a circle, then using the radius of the circle to mark off five equally spaced points along the circumference. These points are then connected to form the pentagon.
Symmetry
A regular pentagon has five lines of symmetry. These lines of symmetry pass through the midpoint of each side and the opposite vertex. The regular pentagon also has a rotational symmetry of order 5, meaning it can be rotated about its center by multiples of 72 degrees and still look the same.
Irregular Pentagon
An irregular pentagon is a pentagon in which not all sides and angles are equal. Irregular pentagons can take on a wide variety of shapes, but they always have five sides and five vertices, and the sum of their interior angles is always 540 degrees.
Applications and Examples
Pentagons, particularly regular pentagons, are commonly found in a variety of practical and theoretical contexts. For example, the regular pentagon is the basis for the shape of the pentagram, a five-pointed star that is used in a variety of cultural and religious contexts. Regular pentagons are also found in the natural world, such as in the shape of the cells of a honeycomb constructed by bees.
In the field of architecture, pentagons are often used in the design of buildings and structures. The most famous of these is perhaps the Pentagon building in Arlington, Virginia, which serves as the headquarters of the United States Department of Defense.
In mathematics, pentagons are often used in the study of geometry and are a key element in the field of polygonal number theory.