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Exterior Angles of Regular Polygons
In the fascinating world of geometry, polygons hold a special place. These closed figures, formed by straight line segments, come in various shapes and sizes. One intriguing aspect of polygons is their exterior angles. This article will delve into the concept of exterior angles, specifically focusing on regular polygons, those with all sides and angles equal.
What are Exterior Angles?
An exterior angle of a polygon is formed by extending one of its sides and measuring the angle between the extended side and the adjacent side. Imagine you have a triangle; if you extend one of its sides, the angle formed outside the triangle is an exterior angle.
Key Properties of Exterior Angles
- Supplementary Angles: Each exterior angle and its corresponding interior angle are supplementary, meaning they add up to 180 degrees.
- Sum of Exterior Angles: The sum of all exterior angles of any polygon, whether regular or irregular, is always 360 degrees.
Exterior Angles of Regular Polygons
In a regular polygon, all sides and angles are equal. This property simplifies the calculation of exterior angles. Here's how:
1. Calculate the Measure of Each Interior Angle:
- Use the formula: Interior Angle = (n-2) * 180 / n, where 'n' is the number of sides of the polygon.
2. Calculate the Measure of Each Exterior Angle:
- Since the interior and exterior angles are supplementary, subtract the interior angle from 180 degrees.
- Exterior Angle = 180 - Interior Angle
Example:
Let's consider a regular hexagon (6 sides).
- Interior Angle: (6-2) * 180 / 6 = 120 degrees
- Exterior Angle: 180 - 120 = 60 degrees
Therefore, each exterior angle of a regular hexagon measures 60 degrees.
Important Note:
The formula for exterior angles applies only to regular polygons. For irregular polygons, each exterior angle needs to be calculated individually.
Visual Representation
The image above shows an exterior angle of a triangle. Notice how it forms a supplementary angle with the interior angle.
Applications of Exterior Angles
Understanding exterior angles is crucial in various geometric problems and applications, including:
- Calculating the number of sides of a regular polygon: Knowing the measure of an exterior angle allows you to determine the number of sides.
- Tessellations: Exterior angles play a vital role in creating tessellations, patterns where shapes fit together without gaps or overlaps.
- Navigation: In navigation, exterior angles are used to calculate bearings and directions.
Conclusion
Exterior angles provide a fascinating way to explore the properties of polygons, particularly regular polygons. By understanding the relationship between interior and exterior angles and the formula for calculating them, you can unlock a deeper appreciation for the beauty and complexity of geometry.