Regular polygon

A regular polygon is a polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

 that is equiangular
Equiangular polygon
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon.The only equiangular triangle is the equilateral triangle...

 (all angles are equal in measure) and equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

 (all sides have the same length). Regular polygons may be convex or star.
These properties apply to all regular polygons, whether convex or star.

A regular n-sided polygon has rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

 of order n.

All vertices of a regular polygon lie on a common circle (the circumscribed circle
Circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....

), i.e., they are concyclic points
Concyclic points
In geometry, a set of points is said to be concyclic if they lie on a common circle.A circle can be drawn around any triangle. A quadrilateral that can be inscribed inside a circle is said to be a cyclic quadrilateral....


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 mid-point.

A regular n-sided polygon can be constructed with compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....

 if and only if the odd prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 factors of n are distinct Fermat primes.