Poligon



Nama dan jenis 
 
Poligon adalah dinamakan mengikut pada jumlah tepi, bergabung satu yunani-terbitan awalan angka dengan akhiran -gon. Contoh pentagon, dodekagon. Segi tiga, sisi empat, dan nonagon adalah pengecualian-pengecualian. Untuk nombor-nombor lebih besar, ahli matematik menulis angka sendiri, contoh 17-gon. Satu variabel boleh juga digunakan, biasanya n gon. Ini adalah jika jumlah berguna bagi tepi adalah digunakan dalam satu formula.

Penamaan poligon 
Poligon yang mempunyai sisi lebih daripada 20 sisi dan kurang daripada 100 sisi dinamakan dengan menggunakan gabungan kata nama berikut:

Contohnya, untuk poligon bersisi 42 akan dinamakan seperti berikut:

dan untuk objek bersisi 50

Namun begitu, poligon yang melebihi nonagons dan decagons, pakar matematik lebih gemar menggunakan aforementioned numeral notation (contohnya, MathWorld mempunyai artikel berkenaan 17-gons dan 257-gons).

Klasifikasi taxonomi 
Taksonomi klasifikasi poligon ditunjukkan melalui gambarajah di bawah:

                                       Polygon
                                      /       \
                                  Simple     Complex
                                 /      \      /
                            Convex    Concave /
                             /    \     /    /
                        Cyclic    Equilateral
                            \     /    
                            Regular

 A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); otherwise it is called complex.
 A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave or non-convex.
 A simple polygon is called equilateral if all edges are of the same length. (A 5 or more sided polygon can be concave and equilateral)
 A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
 A cyclic and equilateral polygon is called regular; for each number of sides, all regular polygons with the same number of sides are similar.
 A simple polygon may also be defined as regular if it is cyclic and equilateral.

Ciri-ciri 
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.

Sudut 
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 a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)

 Any simple n-gon can be considered to be made up of (n−2) triangles, each of which has an angle sum of π radians or 180°.

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).

Keluasan 
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.

Concyclic 
All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).

Titik dalam ujian polygon 
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.

Kotak istimewa 
Some special cases are:

 Angle of 0° or 180° (degenerate case)
 Two non-adjacent sides are on the same line
 Equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
 Equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)

A triangle is equilateral iff it is equiangular.

An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle.

Pautan luar 
 Mathworld: Polygon
 Draw n Polygons Applet to draw a polygon with n vertices.
  More information

Lihat juga 
 Cyclic polygon
 Geometric shape
 Polygon triangulation
 Polyform
 Polyhedron
 Polytope
 Regular polygon
 Simple polygon
 Star polygon
 Synthetic geometry
 Tiling
 Tiling puzzle
 Golygon

Matematik