Polygon

Polygon
Name	Polygon
Caption	A regular polygon, specifically a pentagon
Angle	(n-2) × 180° / n (for regular)
Dual	Self-dual
Properties	Planar, simple, convex or concave

Polygon. In Euclidean geometry, a polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain. The bounded plane region, the bounding circuit, or the two together, may be called a polygon, with the segments as its edges or sides and the points where two edges meet as its vertices or corners. The study of polygons dates back to ancient mathematicians like Euclid and is fundamental to fields ranging from computer graphics to cartography.

Definition and basic properties

A polygon is defined as a two-dimensional geometric figure formed from a finite number of non-collinear points called vertices, connected in sequence by straight line segments called sides, with each vertex shared by exactly two sides. The most fundamental property is that it is a simple polygon, meaning its sides intersect only at the vertices, forming a single closed boundary without self-intersection. Polygons that are not simple are termed complex or self-intersecting, such as the pentagram. The interior of a simple polygon is a single contiguous region, and the polygon together with its interior is called a polygonal region. Key foundational work on these properties is found in Euclid's Elements, particularly in Book I and Book VI. The Jordan curve theorem provides a deeper topological justification for the distinction between interior and exterior regions of a simple closed curve.

Classification of polygons

Polygons are classified primarily by the number of sides, or their order, leading to specific names like triangle (3), quadrilateral (4), pentagon (5), hexagon (6), and heptagon (7). Beyond generic names, they are categorized by their shape properties. A convex polygon has all interior angles less than 180° and every line segment between two points inside the polygon lies entirely within it, exemplified by a square or a regular hexagon. In contrast, a concave polygon (or non-convex) has at least one interior angle greater than 180° and at least one diagonal lies outside the figure. Further classifications include equilateral polygons, where all sides are equal, and equiangular polygons, where all angles are equal; a polygon that is both is regular. Specialized types include the cyclic polygon, where all vertices lie on a single circle, and the tangential polygon, where all sides are tangent to an inscribed circle.

Angles and perimeter

The sum of the interior angles of a simple polygon with n sides is given by the formula (n – 2) × 180 degrees, a result proven by dividing the polygon into triangles from a single vertex. For a regular polygon, each interior angle is equal to this sum divided by n. The sum of the exterior angles, one at each vertex, is always 360 degrees, regardless of the number of sides, a property utilized in surveying and navigation. The perimeter of a polygon is the total length of its boundary, calculated as the sum of the lengths of all its sides. In trigonometry, the Law of sines and Law of cosines are often applied to find unknown side lengths or angles in non-regular polygons, especially triangles. The concept of perimeter is critical in practical applications like fencing a plot of land or determining the circumference of an approximating polygon for a circle.

Area formulas

Calculating the area of a polygon is a central problem in geometry. For a triangle, the basic formula is half the product of its base and height, with alternatives like Heron's formula for sides a, b, c. The area of a rectangle is length times width, and for a parallelogram, it is base times height. The area of any simple polygon can be computed using the Shoelace formula, which employs the coordinates of its vertices, a method foundational in computational geometry and GIS software. For a regular polygon with side length s and apothem a (the distance from the center to a side), the area is (1/2) × perimeter × apothem. The Pick's theorem provides an elegant formula for the area of a simple polygon whose vertices lie on lattice points of a Cartesian coordinate system.

Symmetry and regular polygons

Regular polygons are highly symmetric, possessing both rotational symmetry and reflection symmetry. The dihedral group D_n describes the full symmetry group of a regular n-gon, consisting of n rotations and n reflections. These polygons are both cyclic and tangential, and their vertices lie on a circumscribed circle while their sides are tangent to an inscribed circle. The study of regular polygons is deeply intertwined with constructible polygons in compass and straightedge constructions, a field advanced by Carl Friedrich Gauss, who showed which regular polygons are constructible, such as the heptadecagon. Regular polygons serve as the faces of Platonic solids, like the cube (squares) and dodecahedron (pentagons), and appear in art and architecture, from the Pentagon to the designs of Alhambra.

Polygons in complex geometry

In more advanced geometric settings, the concept of a polygon extends beyond the Euclidean plane. In spherical geometry, a spherical polygon is formed by arcs of great circles on a sphere, with angle sums exceeding those of planar polygons, relevant to astronomy and geodesy. In hyperbolic geometry, discovered independently by Nikolai Lobachevsky and János Bolyai, hyperbolic polygons have angle sums less than their Euclidean counterparts. The theory of complex polygons, or star polygons like the pentagram, involves configurations where edges intersect. In algebraic geometry, polygons appear as Newton polygons in the study of polynomial roots and in toric geometry, where they correspond to algebraic varieties. The generalization to polyhedra in three dimensions and polytopes in higher dimensions is a major theme in the work of mathematicians like H. S. M. Coxeter.

Category:Polygons Category:Elementary shapes Category:Euclidean plane geometry