Difference between revisions of "Polygon"

(Angles in Regular Polygons: added more to table, fixed markup problems)
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== Angles in Regular Polygons ==
 
== Angles in Regular Polygons ==
 
===Exterior===
 
===Exterior===
In any regular polygon, the sum of the exterior angle is equal to <math>360^\circ</math>.  
+
In any regular polygon, the sum of the [[exterior angle]] is equal to <math>360^\circ</math>.
 +
 
 
===Interior===
 
===Interior===
 
The sum of interior angles can be given by the formula <math>180(n-2)^\circ</math>, where<math> n</math> is the number of sides. Thus any angle is <math>\frac{180(n-2)}{n}^\circ</math>.
 
The sum of interior angles can be given by the formula <math>180(n-2)^\circ</math>, where<math> n</math> is the number of sides. Thus any angle is <math>\frac{180(n-2)}{n}^\circ</math>.

Revision as of 17:39, 8 March 2014

A polygon is a closed planar figure consisting of straight line segments. There are two types of polygons: convex and concave.

In their most general form, polygons are an ordered set of vertices, $\{A_1, A_2, \ldots, A_n\}$, $n \geq 3$, with edges $\{\overline{A_1A_2}, \overline{A_2A_3}, \ldots, \overline{A_nA_1}\}$ joining consecutive vertices. Most frequently, one deals with simple polygons in which no two edges are allowed to intersect. (In fact, the adjective "simple" is almost always omitted, so the term "polygon" should be understood to mean "simple polygon" unless otherwise noted.)

A degenerate polygon is one in which some vertex lies on an edge joining two other vertices. This can happen in one of two ways: either the vertices $A_{i - 1},A_i$ and $A_{i+1}$ can be colinear or the vertices $A_i$ and $A_{i + 1}$ can overlap (fail to be distinct). In either of these cases, our polygon of $n$ vertices will appear to have $n - 1$ or fewer -- it will have "degenerated" from an $n$-gon to an $(n - 1)$-gon. (In the case of triangles, this will result in either a line segment or a point.)


Angles in Regular Polygons

Exterior

In any regular polygon, the sum of the exterior angle is equal to $360^\circ$.

Interior

The sum of interior angles can be given by the formula $180(n-2)^\circ$, where$n$ is the number of sides. Thus any angle is $\frac{180(n-2)}{n}^\circ$.

Number of Sides Sum of Interior angles Individual angle measure in regular polygon
3 $180^\circ$ $60^\circ$
4 $360^\circ$ $90^\circ$
5 $540^\circ$ $108^\circ$
6 $720^\circ$ $120^\circ$
8 $1080^\circ$ $135^\circ$

See also

This article is a stub. Help us out by expanding it.

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