**Introduction to Polygon Rules**

All plane polygons share certain things in common. It’s possible to calculate the perimeter or area of any polygon. Certain rules and definitions apply concerning the interior and exterior angles, and the relationships between the angles and the sides. Some of the more significant rules of “polygony” (pronounced “pa-LIG-ah-nee”), a make-believe term that means “the science of polygons,” follow.

**It’s Greek To Us**

**Theta**

Mathematicians, scientists, and engineers often use Greek letters to represent geometric angles. The most common symbol for this purpose is an italicized, lowercase Greek letter theta (pronounced “THAY-tuh”). It looks like a numeral zero leaning to the right, with a horizontal line across its middle ( *θ* ).

**Phi**

When writing about two different angles, a second Greek letter is used along with *θ* . Most often, it is the italicized, lowercase letter phi (pronounced “fie” or “fee”). It looks like a lowercase English letter o leaning to the right, with a forward slash through it ( *φ* ). You might as well get used to these symbols, because if you have anything to do with engineering and science, you’re going to encounter them.

**Alpha, Beta, and Gamma**

Sometimes the italic, lowercase Greek alpha (“AL-fuh”), beta (“BAY-tuh”), and gamma (“GAM-uh”) are used to represent angles. These, respectively, look like this: *α* , *β* , *γ* . When things get messy and there are a lot of angles to talk about, numeric subscripts are sometimes used, so don’t be surprised if you see angles denoted *θ* _{1} , *θ* _{2} , *θ* _{3} , and so on.

**Sum Of Interior Angles**

Let *V* be a plane polygon having *n* sides. Let the interior angles be *θ* _{1} , *θ* _{2} , *θ* _{3} , . . ., *θ* _{n} (Fig. 4-5). The following equation holds if the angular measures are given in degrees:

*θ* _{1} + *θ* _{2} + *θ* _{3} + . . . + *θ* _{n} = 180 *n* − 360 = 180( *n* − 2)

If the angular measures are given in radians, then the following holds:

*θ* _{1} + *θ* _{2} + *θ* _{3} + . . . + *θ* _{n} = *πn* − 2 *π* = *π* ( *n* − 2)

In these examples, the degree symbol (°) and the radian abbreviation (rad) are left out for simplicity. It is all right to do this, as long as it is clear which angular units we’re dealing with.

**Fig. 4-5** . Adding up the measures of the interior angles of a general, many-sided polygon.

**Individual Interior Angles Of Regular Polygon**

Let *V* be a plane polygon having *n* sides whose interior angles all have equal measure given by *θ* , and whose sides all have equal length given by *s* (Fig. 4-6). Then *V* is a regular polygon, and the measure of each interior angle, *θ* , in degrees is given by the following formula:

*θ* = (180 *n* − 360)/ *n*

If the angular measures are given in radians, then the formula looks like this:

*θ* = ( *πn* − 2 *π* )/ *n*

**Fig. 4-6** . Interior angles of a regular, many-sided polygon.

**Positive And Negative Exterior Angles**

An *exterior angle* of a polygon is measured counterclockwise between a specific side and the extension of a side next to it. An example is shown in Fig. 4-7. If the arc of the angle lies outside the polygon, then the resulting angle *θ* has a measure between, but not including, 0 and 180 degrees. The angle is positive because it is measured “positively counterclockwise”:

0° < *θ* < 180°

If the arc of the angle lies inside the polygon, then the angle is measured clockwise (“negatively counterclockwise”). This results in an angle *φ* with a measure between, but not including, −180 and 0 degrees:

−180° < *φ* < 0°

**Fig. 4-7** . Exterior angles of an irregular polygon. The angle *θ* is measured “positively counterclockwise” while the angle *φ* is measured “negatively counterclockwise.”

**Perimeter Of Regular Polygon**

Let *V* be a regular plane polygon having *n* sides of length *s* , and whose vertices are *P* _{1} , *P* _{2} , *P* _{3} , . . ., *P* _{n} as shown in Fig. 4-8. Then the perimeter, *B* , of the polygon is given by the following formula:

*B* = *ns*

**Fig. 4-8** . Perimeter and area of a regular, *n* -sided polygon. Vertices are labeled *P* _{1} , *P* _{2} , *P* _{3} , . . ., *P* _{n} . The length of each side is *s* units.

Practice problems for these concepts can be found at: Other Plane Figures Practice Test.

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From Geometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.