Polygon
A plane figure of connected line segments forming a closed chain.
A polygon is a flat shape made from straight line segments that join up to create a closed loop. These segments are called edges or sides, and the points where two edges meet are called vertices or corners. A polygon with \(n\) sides is known as an \(n\)-gon—for instance, a triangle is a 3-gon.
A simple polygon does not cross itself; the only places its edges meet are at the shared endpoints of consecutive segments. Such a polygon forms the boundary of a region called a solid polygon, whose interior is its body (also referred to as a polygonal region or area). In contexts dealing only with simple and solid polygons, the term "polygon" may refer just to a simple polygon or to a solid one. A polygonal chain can also cross over itself, creating star polygons and other self-intersecting shapes. Some sources consider closed polygonal chains in Euclidean space that do not lie in a single plane to be a type of polygon (a skew polygon). As a 2‑dimensional shape, a polygon is a specific case of the more general polytope, which exists in any number of dimensions.
**Etymology** The word "polygon" comes from the Greek *polús* ("much" or "many") and *gōnía* ("corner" or "angle"), meaning "having multiple angles." It has been suggested that *gónu* ("knee") may be the origin of *gon*.
**Classification**
*Number of sides* Polygons are primarily grouped by how many sides they have.
*Convexity and intersection* Polygons can be described by their convexity or type of non‑convexity: - **Convex**: any line drawn through the polygon (not tangent to an edge or corner) meets its boundary exactly twice. All interior angles are less than 180°, and any line segment with endpoints on the boundary passes only through interior points between them. This holds in any geometry, not just Euclidean. - **Non‑convex**: a line can be found that meets the boundary more than twice, meaning there exists a line segment between two boundary points that goes outside the polygon. - **Simple**: the boundary does not cross itself. All convex polygons are simple. - **Concave**: non‑convex and simple, with at least one interior angle greater than 180°. - **Star‑shaped**: the entire interior is visible from at least one point without crossing any edge. The polygon must be simple and may be convex or concave; all convex polygons are star‑shaped. - **Self‑intersecting**: the boundary
- field
- Geometry
- known_for
- Plane figure made of line segments forming a closed chain; classification by sides, convexity, and symmetry
Lore & Background
The word polygon derives from the Greek adjective πολύς (polús) meaning 'much' or 'many' and γωνία (gōnía) meaning 'corner' or 'angle', thus 'having multiple angles'. It has been suggested that γόνυ (gónu) 'knee' may be the origin of gon. Polygons are primarily classified by the number of sides, and may be characterized by convexity or type of non-convexity, including convex, non-convex, simple, concave, star-shaped, self-intersecting, and star polygon types. Equality and symmetry classifications include equiangular, equilateral, regular, cyclic, tangential, isogonal, and isotoxal.
Reader's Guide
Polygons are central to geometry, providing the basis for understanding shapes, area, and angles. The interior angle sum of a simple n-gon is (n − 2) × 180 degrees, derived from triangulation. The signed area of a simple polygon can be computed using the shoelace formula. Polygons also generalize to higher dimensions as polytopes. Their classification by convexity and symmetry aids in both theoretical and applied mathematics, from computer graphics to architectural design.
Did You Know?
- A simple polygon does not intersect itself; the only allowed intersections are shared endpoints of consecutive segments.
- The sum of the interior angles of a simple n-gon is (n − 2) × 180 degrees.
- A regular polygon is both equilateral and equiangular.
- A star polygon self-intersects in a regular way and cannot be both a star and star-shaped.
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