# The Many Facets of a Polygon

November 2, 2016
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A Polygon is a plane figure formed by at least three sides and angles. The table below displays details of some of the common polygons.

 Name No. of sides No. of angles Triangle 3 3 Quadrilaterals 4 4 Pentagon 5 5 Hexagon 6 6 Heptagon 7 7 Octagon 8 8 Nonagon 9 9 Decagon 10 10 n-gon n n

Polygon Angles: Angles of a polygon are represented in two ways.

Interior Angles: An interior angle of a polygon is an angle formed inside the polygon by any two adjacent sides of the polygon. The sum of the interior angles of a polygon is based on the number of sides the polygon has, and it is calculated as

Sum of interior angles = (n-2) * 180o

“n” is the number of sides of a polygon

Example 1: Calculate the sum of interior angles of a pentagon.

Solution: Pentagon has 5 sides (n =5)

Sum of interior angles = (5-2) * 180o

= 3 * 180o

= 540o

Exterior angles: Angles formed between any side and extension of its adjacent side. The sum of exterior angles of a polygon is equal to 360o . The  Figure below represents the exterior angles of a pentagon.

Fig 1: Exterior angles of a pentagon

Measure of angle A+B+C+D+E= 360o

Based on the length of the sides and size of the angles, polygons are classified as Regular and Irregular polygons.

Regular polygon: Each side of a polygon is equal in length, and each angle is equal in size.

Fig 2:  Regular pentagon

Each angle of a regular polygon =   Sum of angles of polygon/Number of sides of a polygon

Example 2: What is the measure of each interior angle of a regular octagon?

Solution: Octagon has 8 sides (n =8)

Sum of interior angles= (8 – 2) * 180o
= 6  * 180o
= 1080o

Each angle of a regular Octagon = 1080o/8

= 135o

Number of sides of a polygon: Let’s say each exterior angle of a regular polygon is xo , the number of sides of a polygon is calculated as

Number of sides   = Sum of exterior angles/xo

Example 3: The measure of one of the exterior angles of a polygon is 20o . Number of sides of a polygon is?

Solution:   Number of sides   = Sum of exterior angles/20o

= 360o/ 20o

= 18

Therefore, the given polygon has 18 sides.

Irregular polygon: Each side and angle of a polygon is unequal in length and size. Example

Fig 3:  irregular pentagon

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