Pentagon

A pentagon is a geometric shape, or polygon, consisting of five straight sides and five angles. It is one of the basic shapes in geometry, after the triangle and the quadrilateral.

When all sides are equal and all angles are equal, it is called a regular pentagon. Otherwise, it is an irregular pentagon, with sides of different lengths.

Properties of angles

The sum of interior angles in any pentagon is 540 degrees. This is derived from the general formula for polygons: (n − 2) × 180, where n is the number of sides. In a regular pentagon, each interior angle is 108 degrees, and each exterior angle is 72 degrees.

Regular pentagon

A regular pentagon can be inscribed in a circle, with its five vertices equally spaced around the circumference. It has five lines of symmetry, each dividing the shape into two identical halves, and can be rotated by 72 degrees while remaining identical to itself. When all the lines connecting the vertices are drawn, a five-pointed star (pentagram) emerges at the center.

Connection to the golden ratio

In a regular pentagon, the ratio between a diagonal and a side is the golden ratio, approximately equal to 1.618. For this reason, the pentagon and pentagram have been important subjects in the history of ancient Greek mathematics.

Uses and examples

Pentagons appear in natural forms, such as certain flowers and the skeletal structure of five-armed sea creatures. They also appear in the structure of a dodecahedron, a three-dimensional solid composed of 12 pentagonal faces. The term 'Pentagon' is also used for the headquarters of the United States Department of Defense, a building whose shape is based on five sides.