Polygon
Definition
A polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon.
Characteristics
The segments of a polygonal circuit 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. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.
Classification
Polygons are primarily classified by the number of sides. Polygons with three sides are known as triangles and those with four sides are called quadrilaterals. Polygons with five, six, and eight sides are known as pentagons, hexagons, and octagons respectively. The general term for any polygon with n sides is an n-gon.
Polygons can also be classified based on their symmetry. Regular polygons have all sides and angles equal, while irregular polygons do not. Convex polygons have all interior angles less than 180 degrees, while concave polygons have one or more interior angles greater than 180 degrees.
Properties
The sum of the interior angles of a polygon is given by the formula (n-2) × 180 degrees, where n is the number of sides. The area of a polygon can be computed in various ways depending on the type of the polygon.
The properties of polygons are the subject of much study in geometry, particularly in the areas of tiling and convexity. Polygons also have applications in computer graphics where they are used to approximate 3D shapes.
Regular Polygons
A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In a convex polygon, all the vertices point outwards, away from the center, while in a star polygon, some vertices point inwards.
Irregular Polygons
An irregular polygon is any polygon that is not regular. They can have sides of any length and each interior angle can be any measure. They can be either convex or concave.
Convex and Concave Polygons
A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards, away from the interior of the shape. Think of it as a 'bulging' polygon.
On the other hand, a concave polygon is defined as a polygon with one or more interior angles greater than 180°. This means that at least one vertex of the polygon will point inwards towards the interior of the shape. Think of it as a 'caved in' polygon.
Applications
Polygons have many applications in the real world, from architecture and engineering to art and nature. In computer graphics, polygons are used to create 3D models and scenes. In architecture, polygons are used in the design of structures and buildings. In nature, polygons can be observed in the structure of crystals and honeycombs.