Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star. In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed.

General properties

These properties apply to all regular polygons, whether convex or star.

A regular n-sided polygon has rotational symmetry of order n.

All vertices of a regular polygon lie on a common circle (the circumscribed circle), i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon.

A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

Star polygon

In geometry, a star polygon (not to be confused with a star-shaped polygon) is a concave polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined.

Branko Grünbaum identified two primary definitions used by Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and a second are simple isotoxal concave polygons.

The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram.

Etymology

Star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. For example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή (grammḗ) meaning a line.

Regular star polygon

A "regular star polygon" is a self-intersecting, equilateral equiangular polygon, created by connecting one vertex of a simple, regular, p-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. Alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the first to the third vertex, from the third vertex to the fifth vertex, from the fifth vertex to the second vertex, from the second vertex to the fourth vertex, and from the fourth vertex to the first vertex.