Ciri-ciri Poligon

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).