Understanding Regular Polygons: Solving for the Number of Sides Based on Exterior Angle

In the realm of geometry, a polygon is a closed plane figure formed by three or more straight line segments. A regular polygon is a special type of polygon where all sides are equal in length and all interior angles are equal in measure. One characteristic of a regular polygon that simplifies the calculation of its properties is the sum of its exterior angles, which is always 360deg;.

The Significance of Exterior Angles in Regular Polygons

For any polygon, the sum of the exterior angles is always 360deg;. This is a crucial piece of information when dealing with regular polygons. In a regular polygon, each exterior angle is equal. Therefore, the number of sides of a regular polygon can be easily calculated using the formula:

Number of sides 360deg;/Exterior Angle

Solving for the Number of Sides Given an Exterior Angle

Let's consider an example where the measure of each exterior angle of a regular polygon is 40deg;. Using the formula mentioned above, we can calculate the number of sides of this polygon.

Calculation Steps

Sum of exterior angles 360deg; Measure of each exterior angle 40deg; Number of sides 360deg;/40deg; 9 Therefore, the polygon is a nonagon, which has 9 sides.

From this calculation, we can deduce that each interior angle of a nonagon measures 140deg; (since the sum of the interior angles of a polygon is (n-2) x 180deg;, where n is the number of sides). This is calculated as follows:

Interior angle 180deg; - 40deg; 140deg;

Application and Validation

Let's validate this formula with a few more examples:

Example 1: Exterior Angle 44deg;

Sum of exterior angles 360deg; Measure of each exterior angle 44deg; Number of sides 360deg;/44deg; 8.18 Since the number of sides should be an integer, 8.18 is not a valid number of sides for a regular polygon.

In this case, we conclude that there is no regular polygon with an exterior angle of 44deg;. Therefore, the figure in question is an irregular polygon.

Example 2: Exterior Angle 45deg;

Sum of exterior angles 360deg; Measure of each exterior angle 45deg; Number of sides 360deg;/45deg; 8

Here, we have a regular octagon, which has 8 sides and each exterior angle of 45deg;.

Example 3: Exterior Angle 40deg; Revisited

Sum of exterior angles 360deg; Measure of each exterior angle 40deg; Number of sides 360deg;/40deg; 9

We have already determined that a regular polygon with an exterior angle of 40deg; is a nonagon, meaning it has 9 sides.

Conclusion

Understanding the relationship between the exterior angle and the number of sides of a regular polygon is crucial in geometry. By leveraging the fact that the sum of the exterior angles of any polygon is 360deg;, we can easily calculate the number of sides of a regular polygon. This knowledge is valuable for various applications in geometry, engineering, and architecture.

Keywords: Exterior angle, regular polygon, number of sides