A regular polygon is a planar figure whose sides are all the same length and whose interior angles are all equal. In plain terms, it is both equilateral (all edges equal) and equiangular (all angles equal). Every regular polygon has the same number of edges and vertices; that number is often denoted by n. Convex regular polygons include familiar examples such as the equilateral triangle (n=3) and the square (n=4). Self-intersecting regular figures, known as regular star polygons, are also regular in the sense of equal side lengths and equal vertex angles but are nonconvex.
Key properties
Convex regular n-gons are cyclic and tangential: all vertices lie on a common circumscribed circle and all sides are tangent to a common inscribed circle. The center of both circles is the polygon's center, which is equidistant from all vertices and from all side midpoints. Important formulae and facts include the interior angle measure, symmetry, and area expressions. The interior angle of a convex regular n-gon equals (n-2)·180°/n. The rotational and reflection symmetries form the dihedral group Dn, giving 2n isometries in the plane. Two common area formulas are A = (1/4) n s^2 cot(π/n) for side length s, and A = (1/2) n R^2 sin(2π/n) for circumradius R.
Construction and algebraic criteria
Regular polygons can often be constructed with straightedge and compass, but not always. A classical result (the Gauss–Wantzel theorem) characterizes which regular n-gons are constructible: n must equal 2^k times a product of distinct Fermat primes. This explains why regular polygons with certain numbers of sides (for example, n = 3, 4, 5, 15, 17, 257, 65537 under the appropriate factors) are compass-and-straightedge constructible, while many others are not. Practical geometric constructions and numerical approximations are used in design and drafting when exact construction is impossible or impractical.
Types and notation
Regular polygons are denoted by their Schläfli symbol {n}. Star polygons are written {n/k}, where k indicates a step used to connect vertices in sequence; these figures are regular but self-intersecting when k>1 and gcd(n,k)=1. When the polygon is convex, it is sometimes called a regular convex polygon; when self-intersecting, it is a regular star polygon. Distinctions are useful: an equilateral polygon that is not equiangular (for example, a rhombus) is not regular, and an equiangular rectangle with unequal adjacent sides is not regular either.
Examples and applications
Common regular polygons appear in art, architecture, tiling, and engineering. Regular triangles, squares, and hexagons tile the plane without gaps; other regular polygons cannot tile the plane alone. Regular polygons are used in wheels, gears (approximate regular shapes), decorative motifs, and in mathematical problems that exploit their symmetry. Star polygons appear in heraldry and design where striking self-intersecting shapes are desired.
Further distinctions and notable facts
- Every convex regular polygon is both cyclic (circumcircle exists) and tangential (incircle exists). See number of sides and radius relations at standard references.
- Interior angle formula and area depend only on n and either side length or radius; see discussions on equal length conventions and formulas.
- Star polygons ({n/k}) illustrate how regularity extends beyond convexity; for more on star forms consult Schläfli notation summaries.
- Symmetry group Dn governs dihedral symmetries; related material appears under dihedral group expositions.
- Classic constructibility rules are treated in number-theoretic contexts; see accounts of the Gauss–Wantzel criterion at constructible polygons.
- To explore examples, diagrams, and further geometric proofs, consult introductory geometry resources at polygon references.
Regular polygons combine simple local rules (equal sides and angles) with rich global structure (circles, symmetries, and algebraic constructibility). Their study connects elementary Euclidean geometry to group theory, number theory, and applications in design and the visual arts.
