A **convex polygon** is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set.^{ [1] } In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.

The following properties of a simple polygon are all equivalent to convexity:

- Every internal angle is strictly less than 180 degrees.
- Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary.
- The polygon is entirely contained in a closed half-plane defined by each of its edges.
- For each edge, the interior points are all on the same side of the line that the edge defines.
- The angle at each vertex contains all other vertices in its edges and interior.
- The polygon is the convex hull of its edges.

Additional properties of convex polygons include:

- The intersection of two convex polygons is a convex polygon.
- A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices.
- Helly's theorem: For every collection of at least three convex polygons: if the intersection of every three of them is nonempty, then the whole collection has a nonempty intersection.
- Krein–Milman theorem: A convex polygon is the convex hull of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon to recover the entire polygon shape.
- Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line. If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines (with a gap between them).
**Inscribed triangle**property: Of all triangles contained in a convex polygon, there exists a triangle with a maximal area whose vertices are all polygon vertices.^{ [2] }**Inscribing triangle**property: every convex polygon with area*A*can be inscribed in a triangle of area at most equal to 2*A*. Equality holds (exclusively) for a parallelogram.^{ [3] }**Inscribed/inscribing rectangles**property: For every convex body C in the plane, we can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and .^{ [4] }- The mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the same perimeter as the polygon.
^{ [5] }

Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not self-intersecting, is convex. However, not every convex polygon can be inscribed in a circle.

The following properties of a simple polygon are all equivalent to strict convexity:

- Every internal angle is strictly less than 180 degrees.
- Every line segment between two points in the interior, or between two points on the boundary but not on the same edge, is strictly interior to the polygon (except at its endpoints if they are on the edges).
- For each edge, the interior points and the boundary points not contained in the edge are on the same side of the line that the edge defines.
- The angle at each vertex contains all other vertices in its interior (except the given vertex and the two adjacent vertices).

Every nondegenerate triangle is strictly convex.

- Concave polygon a simple polygon that is not convex
- Convex polytope
- Cyclic polygon
- Implicit curve § Smooth approximation of convex polygons
- Tangential polygon

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

In 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 or *polygonal circuit*. The solid plane region, the bounding circuit, or the two together, may be called a **polygon**.

In Euclidean plane geometry, a **quadrilateral** is a polygon with four edges (sides) and four vertices (corners). Other names for quadrilateral include **quadrangle**, **tetragon**, and **4-gon**. A quadrilateral with vertices , , and is sometimes denoted as .

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

In Euclidean plane geometry, a **rectangle** is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term **oblong** is occasionally used to refer to a non-square rectangle. A rectangle with vertices *ABCD* would be denoted as *ABCD*.

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In geometry, **bisection** is the division of something into two equal or congruent parts, usually by a line, which is then called a *bisector*. The most often considered types of bisectors are the *segment bisector* and the *angle bisector*.

In plane Euclidean geometry, a **rhombus** is a quadrilateral whose four sides all have the same length. Another name is **equilateral quadrilateral**, since equilateral means that all of its sides are equal in length. The rhombus is often called a **diamond**, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a **lozenge**, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, **Pick's theorem** provides a simple formula for calculating the area A of this polygon in terms of the number i of *lattice points in the interior* located in the polygon and the number b of *lattice points on the boundary* placed on the polygon's perimeter:

In Euclidean geometry, a **regular polygon** is a polygon that is equiangular and equilateral. Regular polygons may be either **convex** or **star**. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

In geometry, a **simple polygon** is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general.

A **convex polytope** is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

**Geometric graph theory** in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs".

The classical **four-vertex theorem** states that the curvature function of a simple, closed, smooth plane curve has at least four local extrema. The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion.

In Euclidean plane geometry, a **pseudotriangle** (*pseudo-triangle*) is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A **pseudotriangulation** (*pseudo-triangulations*) is a partition of a region of the plane into pseudotriangles, and a **pointed pseudotriangulation** is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π.

In geometry, a **vertex**, often denoted by letters such as , , , , is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

In polyhedral combinatorics, a branch of mathematics, **Steinitz's theorem** is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”

In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In complex analysis, the **Schwarz triangle function** or **Schwarz s-function** is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. Let *πα*, *πβ*, and *πγ* be the interior angles at the vertices of the triangle. If any of *α, β*, and *γ* are greater than zero, then the Schwarz triangle function can be given in terms of hypergeometric functions as:

- ↑ Definition and properties of convex polygons with interactive animation.
- ↑ -, Christos. "Is the area of intersection of convex polygons always convex?".
*Math Stack Exchange*.CS1 maint: numeric names: authors list (link) - ↑ Weisstein, Eric W. "Triangle Circumscribing".
*Wolfram Math World*. - ↑ Lassak, M. (1993). "Approximation of convex bodies by rectangles".
*Geometriae Dedicata*.**47**: 111. doi:10.1007/BF01263495. - ↑ Jim Belk. "What's the average width of a convex polygon?".
*Math Stack Exchange*.

Wikimedia Commons has media related to . Convex polygons |

- Weisstein, Eric W. "Convex polygon".
*MathWorld*. - http://www.rustycode.com/tutorials/convex.html
- Schorn, Peter; Fisher, Frederick (1994), "I.2 Testing the convexity of a polygon", in Heckbert, Paul S. (ed.),
*Graphics Gems IV*, Morgan Kaufmann (Academic Press), pp. 7–15, ISBN 9780123361554

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