In geometry, a pentagon (from Greek πέντε (pente) 'five' and γωνία (gonia) 'angle'[1]) is any five-sided polygon or 5-gon.

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

The diagonals of a convex regular pentagon are in the golden ratio to its sides.

(distance between two farthest separated points, which equals the diagonal length

are given by: The area of a convex regular pentagon with side length

the regular pentagon fills approximately 0.7568 of its circumscribed circle.

For an arbitrary point in the plane of a regular pentagon with circumradius

One method to construct a regular pentagon in a given circle is described by Richmond[3] and further discussed in Cromwell's Polyhedra.

[4] The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon.

Its center is located at point C and a midpoint M is marked halfway along its radius.

A horizontal line through Q intersects the circle at point P, and chord PD is the required side of the inscribed pentagon.

To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle.

Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as

The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.

[5] This methodology leads to a procedure for constructing a regular pentagon.

The steps are as follows:[6] Steps 6–8 are equivalent to the following version, shown in the animation: A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge.

John Conway labels these by a letter and group order.

The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices.

Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms.

An equilateral pentagon is a polygon with five sides of equal length.

However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons.

In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal).

A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices.

The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon.

In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has the optimal density among all packings of regular pentagons in the plane.

[16] There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon.

For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent.