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A polygon (IPA: ˈpɒliɡən}}, from Greek, literally "many-angle") is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.
Names and types
Naming polygons To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows In many cases KAI is not necessary. That is, a 42-sided figure would be named as follows: and a 50-sided figure But beyond enneagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). Taxonomic classification The taxonomic classification of polygons is illustrated by the following graph: Properties We will assume Euclidean geometry throughout. An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape. In the case of a line of symmetry the latter reduces to n-2. Let k≥2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n-1 degrees of freedom. Angles Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. Area The area of a polygon is the measurment of the 2-dimensional region enclosed by the polygon. The area A of a simple polygon can be computed if the cartesian coordinates (x1, y1), (x2, y2), ..., (xn, yn) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is A = ½ · (x1y2 − x2y1 + x2y3 − x3y2 + ... + xny1 − x1yn) = ½ · (x1(y2 − yn) + x2(y3 − y1) + x3(y4 − y2) + ... + xn(y1 − yn−1)) The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. 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. If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. For a regular polygon with n sides of length s, the area is given by: In simple terms a polygon is a multi sided shape made of verticies, edges and faces. Point in polygon test In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test. Special cases Some special cases are: A triangle is equilateral if and only if it is equiangular. An equilateral quadrilateral is a rhombus. An equiangular quadrilateral is a rectangle or a complex "angular eight" with vertices on a rectangle. A quadrilateral is a square if and only if it is both equilateral and equiangular. Likewise, a quadrilateral is a square if and only if it is both a rhombus and a rectangle. See also | ||||||||||
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