In geometry, a polygon is a flat shape made up of straight line segments connected to form a closed shape. The line segments that make up the closed shape are called edges or sides. The points where two edges meet are called vertices or corners. A polygon with n sides is called an n-gon. For example, a triangle is a 3-gon.

A simple polygon is a shape that does not cross over itself. This means the only points where the line segments touch are the ends of the segments where they connect. A simple polygon forms the outline of a flat area called a solid polygon. The inside of a solid polygon is called its body, also known as a polygonal region or polygonal area. In some cases, when only simple or solid polygons are being discussed, the word "polygon" may refer to one of these types.

A polygonal chain can sometimes cross over itself, creating shapes like star polygons or other self-intersecting polygons. Some sources also describe closed polygonal chains in 3D space as a type of polygon, called a skew polygon, even if the chain is not flat.

A polygon is a two-dimensional example of a more general shape called a polytope, which can exist in any number of dimensions. There are many other types of polygons defined for different purposes.

Etymology

The word "polygon" comes from two Greek words: πολύς (polús), which means "many" or "much," and γωνία (gōnía), which means "corner" or "angle." Some people think that the Greek word γόνυ (gónu), meaning "knee," might be the source of the ending "gon" in "polygon."

Classification

Polygons are mainly grouped based on how many sides they have.

Polygons can also be described by their shape and how their sides are arranged:

• Convex: Any straight line drawn through the polygon (not touching an edge or corner) will cross the boundary exactly twice. This means all interior angles are less than 180°. A line connecting two points on the edge will only pass through the inside of the polygon. This rule applies to all types of geometry, not just flat (Euclidean) geometry.
• Non-convex: A straight line can cross the boundary more than twice. This means there is at least one line connecting two points on the edge that goes outside the polygon.
• Simple: The sides of the polygon do not cross each other. All convex polygons are simple.
• Concave: A non-convex and simple polygon. At least one interior angle is greater than 180°.
• Star-shaped: Every point inside the polygon can be seen from at least one specific point without crossing any edges. The polygon must be simple and can be convex or concave. All convex polygons are star-shaped.
• Self-intersecting: The sides of the polygon cross each other. The term "complex" is sometimes used to describe this, but it can be confusing because "complex" also refers to polygons in a different type of geometry.
• Star polygon: A polygon that crosses itself in a regular, repeating pattern. A star polygon cannot be both a star and star-shaped.

Other properties of polygons include:

• Equiangular: All angles at the corners are equal.
• Equilateral: All sides are the same length.
• Regular: A polygon that is both equilateral and equiangular.
• Cyclic: All corners lie on a single circle, called the circumcircle.
• Tangential: All sides are touched by a circle inside the polygon.
• Isogonal (vertex-transitive): All corners are in the same position within a symmetrical pattern. These polygons are also cyclic and equiangular.
• Isotoxal (edge-transitive): All sides are in the same position within a symmetrical pattern. These polygons are also equilateral and tangential.

A polygon is regular if it is both isogonal and isotoxal, or if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.

• Rectilinear: All sides meet at right angles (90° or 270°).
• Monotone with respect to a line L: Any line that is perpendicular to line L will cross the polygon no more than twice.

Properties and formulas

Euclidean geometry is used throughout this discussion.

Any polygon has the same number of corners as it has sides. Each corner has multiple angles, but two are most important:

Interior Angle – The total of the interior angles in a simple polygon with n sides is (n − 2) × 180 degrees. This is because a polygon with n sides can be divided into (n − 2) triangles, and each triangle has angles that add up to 180 degrees. For a regular convex polygon with n sides, each interior angle measures (1 − 2/n) × π radians or 180 − 360/n degrees. Regular star polygons, studied by Poinsot, have interior angles calculated as π(p − 2q)/p radians or 180(p − 2q)/p degrees, where p and q describe the shape.

Exterior Angle – The exterior angle is the angle formed outside the polygon at each corner, supplementary to the interior angle. When tracing around a convex polygon, the total of all exterior angles equals 360 degrees. For other polygons, the total can be multiples of 360 degrees, such as 720 degrees for a pentagram or 0 degrees for certain shapes.

The vertices of the polygon are labeled as (x₀, y₀), (x₁, y₁), …, (xₙ₋₁, yₙ₋₁) in order. For some formulas, (xₙ, yₙ) is set equal to (x₀, y₀).

For a simple (non-self-intersecting) polygon, the signed area can be calculated using:
– A formula involving coordinates, or
– A formula using determinants and squared distances between vertices.

The signed area depends on the order of the vertices and the orientation of the plane. If the vertices are listed counterclockwise, the area is positive; otherwise, it is negative. The absolute value of this result gives the actual area, known as the shoelace formula.

The area of a simple polygon can also be calculated if the side lengths and exterior angles are known. This method was described by Lopshits in 1963.

If a polygon is drawn on a grid with vertices at grid points, Pick’s theorem provides a simple formula for the area: the number of interior grid points plus half the number of boundary points, minus 1.

For any polygon with perimeter p and area A, the isoperimetric inequality states that p² > 4πA.

The Bolyai–Gerwien theorem says that any two simple polygons with the same area can be cut into pieces that can be rearranged to form each other.

The side lengths of a polygon alone do not determine its area. However, if the polygon is both simple and cyclic (all vertices lie on a circle), the side lengths do determine the area. Among all n-gons with the same side lengths, the one with the largest area is cyclic. Among all n-gons with the same perimeter, the one with the largest area is regular (and thus cyclic).

Special formulas exist for calculating the area of regular polygons. The area can be expressed using the apothem (the radius of the inscribed circle) and the perimeter as:

The area of a regular n-gon can also be expressed using the circumradius (the radius of the circumscribed circle) as:

For self-intersecting polygons, area can be calculated in two ways:
1. Using formulas for simple polygons, with regions inside the polygon multiplied by a "density" factor (e.g., a pentagram’s center has density 2).
2. Considering the area covered by the polygon as a set of points, treating it as one or more simple shapes (e.g., a cross-quadrilateral is treated as two triangles).

The centroid (center of mass) of a solid simple polygon can be calculated using its vertex coordinates and the signed area A. For triangles, the centroid of the vertices and the shape are the same. For polygons with more than three sides, these centroids differ.

Generalizations

The concept of a polygon has been expanded in several ways. Important examples include:

• A spherical polygon is made up of curved lines called arcs of great circles and points (vertices) on the surface of a sphere. It can include a digon, a shape with only two sides and two corners, which cannot exist on a flat surface. Spherical polygons are important in map making and in a method called Wythoff's construction used to create uniform polyhedra.
• A skew polygon is not flat and moves in three or more dimensions. Well-known examples are the Petrie polygons found in regular polytopes.
• An apeirogon is an endless sequence of sides and angles that does not form a closed shape and has no beginning or end.
• A skew apeirogon is an endless sequence of sides and angles that does not lie in a flat plane.
• A polygon with holes is a shape connected in one or more areas, with one outer edge and one or more inner edges (holes).
• A complex polygon is a shape similar to a regular polygon but exists in a special space with two real and two imaginary dimensions.
• An abstract polygon is a mathematical structure that shows how parts like sides and vertices are connected. A real polygon is considered a version of this abstract structure. Depending on how it is mapped, all these types can be represented.
• A polyhedron is a three-dimensional shape with flat polygonal faces, similar to how a polygon is two-dimensional. In four or more dimensions, these shapes are called polytopes. (Some systems use the term "polytope" for all dimensions, with the rule that a polytope must be bounded.)

Naming

The word "polygon" comes from Latin, which was influenced by Greek. In Greek, the word means "many-angled." Polygons are named based on the number of sides they have. Most names combine a Greek number prefix with the ending "-gon," such as "pentagon" (five sides) or "dodecagon" (12 sides). Exceptions include "triangle" (three sides) and "quadrilateral" (four sides).

For polygons with more than 10 sides, mathematicians often use numbers instead of Greek prefixes. For example, a 17-sided polygon is called a "17-gon," and a 257-sided polygon is called a "257-gon."

Some polygons have special names, like the "pentagram," which is a five-pointed star shape and also a type of regular star pentagon.

To name a polygon with more than 20 sides but fewer than 100, prefixes are combined. The word "kai" is sometimes used in these names, especially for polygons with 13 or more sides. This term was used by the scientist Johannes Kepler and later supported by John H. Conway to help clarify names for certain types of shapes. However, not all sources include "kai" in their naming.

History

Polygons have been known since ancient times. Regular polygons were known to the ancient Greeks. A pentagram, which is a non-convex regular polygon (also called a star polygon), appeared as early as the 7th century B.C. on a krater made by Aristophanes. This krater was found in Caere and is now in the Capitoline Museum.

In the 14th century, Thomas Bradwardine conducted the first known organized study of non-convex polygons in general.

In 1952, Geoffrey Colin Shephard expanded the concept of polygons to the complex plane. In this system, each real dimension is paired with an imaginary one to form complex polygons.

In nature

Polygons can be found in rock formations, often as the flat sides of crystals. The angles between these sides depend on the kind of mineral that makes up the crystal.

Regular hexagons may form when lava cools and creates closely packed basalt columns. These columns can be seen at the Giant's Causeway in Northern Ireland and at the Devil's Postpile in California.

In biology, the surface of a wax honeycomb made by bees has a pattern of hexagons. The sides and bottom of each cell in the honeycomb are also polygons.

Computer graphics

In computer graphics, a polygon is a basic shape used to create and display images. Polygons are stored in a database with information about their points (called vertices), which include coordinates, colors, textures, and how the shape is shaded. They also include details about how the points connect and the materials used for the surface.

To model a surface, computer graphics use a grid of shapes called a polygon mesh. If a square mesh has n + 1 points along each side, the mesh contains n squared squares or 2n squared triangles (since each square can be split into two triangles). Each triangle in the mesh has (n + 1)/2(n) vertices. When n is large, this number gets close to one half. Additionally, each point inside the mesh connects to four lines (edges).

The computer system retrieves the structure of polygons needed to create a scene from the database. This information is sent to active memory and then to the display system (such as a screen or monitor) so the scene can be viewed. During this process, the system adjusts the polygons to appear in the correct three-dimensional position on the screen, even though they are two-dimensional shapes.

In computer graphics and geometry, it is often important to determine if a specific point, P = (x₀, y₀), is inside a simple polygon made of connected line segments. This is known as the point-in-polygon test.