Polygon

Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.

Polygons are primarily classified by the number of sides. See the table below.

Polygons may be characterized by their convexity or type of non-convexity:

Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.

For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.

The area of a self-intersecting polygon can be defined in two different ways, giving different answers:

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are

The idea of a polygon has been generalized in various ways. Some of the more important include:

Polygons appear in rock formations, most commonly as the flat facets of crystals, where the angles between the sides depend on the type of mineral from which the crystal is made.

The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.