Pentagon

In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.

A regular pentagon has Schläfli symbol {5} and interior angles of 108°.

A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length ${\displaystyle t,}$ its height ${\displaystyle H}$ (distance from one side to the opposite vertex), width ${\displaystyle W}$ (distance between two farthest separated points, which equals the diagonal length ${\displaystyle D}$) and circumradius ${\displaystyle R}$ are given by:

The area of a convex regular pentagon with side length ${\displaystyle t}$ is given by

If the circumradius ${\displaystyle R}$ of a regular pentagon is given, its edge length ${\displaystyle t}$ is found by the expression

and its area is

since the area of the circumscribed circle is ${\displaystyle \pi R^{2},}$ the regular pentagon fills approximately 0.7568 of its circumscribed circle.