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Poligon Ciri-ciriWe 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.
Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns). This can be seen in two different ways:
Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics).
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.
All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).
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WikiPedia: Poligon http://www.2dcurves.com/line/linep.html http://www25.brinkster.com/denshade/DrawNthPolygon... http://translate.google.com/translate?u=https://en... http://mathworld.wolfram.com/EquiangularPolygon.ht... http://mathworld.wolfram.com/EquilateralPolygon.ht... http://mathworld.wolfram.com/Hexacontagon.html http://mathworld.wolfram.com/Polygon.html http://mathworld.wolfram.com/Tetradecagon.html http://mathworld.wolfram.com/Tridecagon.html http://prpm.dbp.gov.my/