Polygon
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A polygon is a geometric figure that lies on a plane and is made up of a finite number of straight lines whose endpoints are joined so as to form a closed ring. The lines are called sides (or edges), while the points where two sides meet are called vertices (or corners).
Every polygon is counted by the number of sides it has, which is always equal to the number of vertices it has. The smallest polygon is the triangle, which has three sides; next come the quadrilateral, pentagon, hexagon, and larger figures. There is no polygon with only two sides if the plane being considered is flat.
Categories of polygons
Polygons are classified in several ways. When all interior angles are less than 180 degrees, it is called a convex polygon; if any angle is greater than 180 degrees, it is called a concave polygon. They are also classified as regular polygons, in which all sides are equal and all angles are equal, such as an equilateral triangle or a square. If the sides intersect each other, it is called a self-intersecting polygon, an example being a pentagram.
Angles and calculation
The sum of interior angles in a polygon with n sides equals (n − 2) × 180 degrees. Therefore a triangle has 180 degrees and a quadrilateral has 360 degrees. The sum of exterior angles in a convex polygon is always 360 degrees. A regular polygon with n sides has each interior angle equal to ((n − 2) × 180) / n degrees.
Perimeter and area
The perimeter of a polygon is the sum of the lengths of all its sides. The area can be calculated in various ways: polygons such as triangles and quadrilaterals have specific formulas, while more complex polygons can be divided into triangles whose areas are then summed. When the positions of vertices are known by coordinates, a method known as the shoelace formula can be used.
Applications
Polygons are fundamental to the geometry of construction, land measurement, and engineering surveying. Computer programs for three-dimensional graphics also construct images using many small polygons—usually triangles—joined together, a system known as a polygon mesh. Geographic information systems (GIS) also represent regions and boundaries as polygons.