Polygon

Pronunciation: /ˈpɒl iˌgɒn/ ?

A polygon is an n-sided closed figure in a plane.[1][2] Each of the sides of a polygon is a straight line segment. Since a polygon is closed, each of the line segments is connected to another at both ends. Examples of polygons are triangles, rectangles, and pentagons. An n-sided polygon is called an n-gon.

Parts of a polygon

A irregular pentagon. One of the line segments is labeled 'side'. One of the points where two line segments meet is labeled 'vertex'. Inside the polygon is labeled 'interior'. Outside the polygon is labeled 'exterior'. The angle at one of the vertices on the inside of the polygon is labeled internal angle.
Illustration 1: Parts of a polygon.

  • Each of the line segments that define the polygon is called a side or less commonly, an edge.
  • A point where two sides of the polygon meet is called a vertex.
  • Everything inside the polygon is called the interior.
  • Everything outside the polygon is called the exterior.
  • An angle at a vertex on the inside of the polygon is called an interior angle.

Properties of polygons

  • Straight line segments - The edges of polygons are straight line segments. Each line segment is connected to the other line segments at its endpoints.
  • Closed - Polygons are closed figures. A pencil placed on one point on the edge of the polygon and traced around the edge will return to the same point without retracing.
  • Connected - Polygons are all in one piece.
  • convex or convex - A polygon can be concave or convex. A polygon is convex if a line segment drawn between any two points in the polygon remains within the polygon.
  • Simple or complex - A polygon can be simple or complex. In a simple polygon, none of the edges cross.

Regular polygon

A regular polygon is a polygon that is simple, convex and equilateral.

Types of polygons

NameSidesIllustration--- Regular ----
Central
angle
Internal
angle
Inradius/
apothem
CircumradiusArea
n-gon n A variety of polygons, each having straight sides. alpha = (360 degrees)/n = (2 pi rad.)/n beta = (180 degrees(n-2))/n = (pi rad.(n-2)/n) r1 = (1/2)*s*cot(alpha/2) r2 = (1/2)*s*sin(alpha/2) A = (1/4)*n*s^2*cot(180 degrees/n) = (1/4)*n*s^2*cot(pi rad/n)
triangle 3 A three sided polygon. alpha = 120 degrees = 2/3 pi rad. beta = 60 degrees = pi/3 rad. r1 = (1/2)*s*cot(60 deg) = (1/2)*s*cot(1/3 pi rad.) = 1/(2 square root(3)) s approximately 0.288675s r2 = (1/2)*s*sin(60 deg) = (1/2)*s*sin(1/3 pi rad.) approximately 0.433013s A = (3/4)*s^2*cot(60 degrees) = (3/4)*s^2*cot(pi/3 rad) approximately 0.433013s^2
quadrilateral 4 A four sided polygon. alpha = 90 degrees = 1/2 pi rad. beta = 90 degrees = 1/2 pi rad. r1 = (1/2)*s*cot(45 deg) = (1/2)*s*cot(1/4 pi rad.) = (1/2)s r2 = (1/2)*s*sin(45 deg) = (1/2)*s*sin(1/4 pi rad.) = square root(2)/4 s approximately 0.353553s A = (3/4)*s^2*cot(45 degrees) = (3/4)*s^2*cot(pi/4 rad) = (3/4)s^2
pentagon 5 A five sided polygon. alpha = 72 degrees = 2/5 pi rad. beta = 108 degrees = 3/5 pi rad. r1 = (1/2)*s*cot(36 deg) = (1/2)*s*cot(1/5 pi rad.) approximately 0.688191s r2 = (1/2)*s*sin(36 deg) = (1/2)*s*sin(pi/5 rad.) = approximately 0.293893s A = (3/4)*s^2*cot(36 deg) = (3/4)*s^2*cot(pi/5 rad) approximately 1.03229s
hexagon 6 A six sided polygon. alpha = 60 degrees = 1/3 pi rad. beta = 120 degrees = 2/3 pi rad. r1 = (1/2)*s*cot(30 deg) = (1/2)*s*cot(1/6 pi rad.) = square root(3)/2 s approximately 0.866025s r2 = (1/2)*s*sin(30 deg) = (1/2)*s*sin(1/6 pi rad.) = s/4 A = (3/4)*s^2*cot(30 degrees) = (3/4)*s^2*cot(pi/6 rad) = (3*square root(3)/4)s^2 approximately 1.29904s^2
Table 2: Types of polygons

More Information

  • McAdams, David E. Regular Polygon. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Regular%20Polygon.

Cite this article as:


Polygon. 2009-12-31. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/polygon.html.

Translations

Image Credits

Revision History


2009-12-31: Added "References" (McAdams, David.)
2008-09-03: Expanded 'More Information' (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License