**Convex Polygon** A simple polygon is a finite region of the plane bounded by a non-self-sealing closed broken line. Taking into account how a simple polygon is with respect to a line that contains two of its points, two classes of polygons result: convex polygons and concave polygons.

## Summary

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- 1 Definition
- 1 Examples

- 2 elements
- 1 Topological concepts

- 3 Features
- 4 Sources
- 5 References
- 6 See also

## Definition

A polygon is convex when a line that contains any of its sides determines two half planes, such that in one of them the whole polygon remains; in case no such property occurs the polygon “is not convex”. ^{[one]}

### Examples

- Triangles
- Square, rectangle, rhombus, rhomboid, trapezoid and trapezoids.
- Pentagons, hexagons, octagons, enegons, decagons, hexagons, dodecagons, etc.

## Elements

- Side of the polygon is any segment of the broken line
- Vertex any end of one side, this vertex on two sides called
*adjacent* - Diagonal: Any segment that joins two non-consecutive vertices.
- From the vertex of an entering angle, a decomposition into convex figures with a smaller number of sides is possible, if possible into triangles.

### Topological concepts

- Inner point of a convex polygon. If we draw any line through point I that cuts two sides of the polygon at M and N respectively, if I is between M and N it is
*an interior point*of the polygon. - Interior: It is the set of all interior points.
- Polygonal region: It is the union of the polygon and its interior.
- Border is the broken line that delimits the polygon.
- Exterior point: It is a point that is not in the polygonal region. The set of all exterior points is the
*exterior*of the polygon. - The interior, exterior and polygon establish a plane partition, so that the intersection of two of them is empty and the union of the three sets is equal to the entire plane.
- Accumulation point of a polygon Π is a point P
_{a}that has an environment V (P_{a}; δ) so that it contains at least one point Π other than P_{a}. - Isolated point of a polygon K is a point P
_{i}that is in K and that has an environment V (P_{i}; δ), which does not contain any point of K, outside it.

## characteristics

- In a convex polygon all its diagonals are on the same polygon; otherwise they are inside except for the ends that are on the border.
- Any two different points on the polygon determine a segment that is on the polygon.
- If a line is drawn through an interior point, it is cut at least two of its sides.
- Three-sided polygons (
*triangles*) are the only polygons that cannot be concave, since none of their three angles can exceed 180 degrees or π radians.