# Regular Polygon

9248 Words Oct 13th, 2012 37 Pages
Polygon
For other uses, see Polygon (disambiguation).

Some polygons of different kinds
In geometry a polygon ( /ˈpɒlɪɡɒn/) is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.
A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general
The polygon is also equilateral. * Tangential: all sides are tangent to an inscribed circle. * Regular: A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.
Miscellaneous
* Rectilinear: a polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees. * Monotone with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice.
Properties
Euclidean geometry is assumed throughout.
Angles
Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are: * Interior angle – The sum of the interior angles of a simple n-gon is (n − 2)π radians or (n − 2)180 degrees. This is because 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 degrees. The measure of any interior angle of a convex regular n-gon is radians or degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra. * Exterior angle – Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be