How To Find The Measure Of Each Exterior Angle

How to Find the Measure of Each Exterior Angle: A Comprehensive Guide

Understanding exterior angles is crucial in geometry, particularly when dealing with polygons. This comprehensive guide will walk you through various methods to find the measure of each exterior angle, catering to different levels of understanding and polygon types. We'll explore the relationship between interior and exterior angles, delve into formulas, and provide step-by-step examples to solidify your grasp of this concept.

Understanding Exterior Angles

Before we jump into the calculations, let's define what an exterior angle is. An exterior angle of a polygon is formed by extending one of its sides. It's the angle between the extended side and the adjacent side of the polygon. Each vertex of a polygon has two exterior angles, one on each side, but we typically focus on one exterior angle per vertex. Importantly, an exterior angle and its adjacent interior angle are supplementary—they add up to 180 degrees.

Key Relationship: Interior Angle + Exterior Angle = 180°

This relationship is fundamental to solving problems involving exterior angles. If you know the measure of the interior angle, you can easily calculate the exterior angle, and vice-versa.

Methods for Finding Exterior Angles

The method for finding the measure of each exterior angle depends on the type of polygon and the information provided. Let's explore the most common scenarios:

1. Using the Sum of Exterior Angles

This is the most straightforward method, applicable to all convex polygons (polygons where all interior angles are less than 180°). The sum of the exterior angles of any convex polygon is always 360 degrees. This holds true regardless of the number of sides.

Formula: Sum of Exterior Angles = 360°

How to use this method:

1. Regular Polygons: If the polygon is regular (all sides and angles are equal), simply divide the sum of exterior angles (360°) by the number of sides (n) to find the measure of each exterior angle.

   Formula for Regular Polygons: Each Exterior Angle = 360°/n

   Example: Find the measure of each exterior angle of a regular hexagon (6 sides).

   Each Exterior Angle = 360°/6 = 60°

2. Irregular Polygons: For irregular polygons, you need more information. You might be given the measures of some interior angles or exterior angles. You would then use the supplementary relationship (interior + exterior = 180°) to find any missing exterior angles, ensuring their sum is 360°.

   Example: A quadrilateral has exterior angles of 70°, 90°, and 110°. Find the measure of the fourth exterior angle.

   360° - (70° + 90° + 110°) = 90°

   The fourth exterior angle measures 90°.

2. Using Interior Angles

If you know the measure of each interior angle, you can readily calculate the corresponding exterior angle using the supplementary relationship.

Formula: Exterior Angle = 180° - Interior Angle

How to use this method:

1. Regular Polygons: First, calculate the sum of the interior angles using the formula (n-2) * 180°, where 'n' is the number of sides. Then, divide by the number of sides to find the measure of each interior angle. Finally, subtract this value from 180° to get the exterior angle.

   Example: Find the measure of each exterior angle of a regular pentagon (5 sides).

   Sum of Interior Angles = (5-2) * 180° = 540° Each Interior Angle = 540°/5 = 108° Each Exterior Angle = 180° - 108° = 72°

2. Irregular Polygons: If you know the measure of each interior angle, simply apply the formula (180° - Interior Angle) to each angle individually to find its corresponding exterior angle.

3. Using the Properties of Triangles

This method is particularly useful when dealing with problems involving triangles within a larger polygon. Remember that the sum of angles in any triangle is always 180°. By dissecting the polygon into triangles, you can find the relationships between interior and exterior angles.

Example: Consider a regular pentagon. We can divide it into three triangles by drawing lines from one vertex to the other non-adjacent vertices. Each triangle has angles that add up to 180°. By analyzing the angles of these triangles, we can deduce the relationship and then the measure of each exterior angle.

Advanced Concepts and Problem Solving

1. Non-Convex Polygons

The sum of exterior angles is only 360° for convex polygons. For non-convex polygons (polygons with at least one interior angle greater than 180°), you'll need a different approach. You'll need to carefully consider the orientation of the exterior angles and the angles might not all be directed outwards. It is often easier to find the interior angles first and then calculate the exterior angles using the supplementary relationship.

2. Complex Polygon Decomposition

For highly irregular polygons, it can be beneficial to decompose (break down) the polygon into simpler shapes like triangles and quadrilaterals. This allows for applying simpler geometric principles to solve for individual angles and subsequently, the exterior angles.

3. Using Coordinate Geometry

In some cases, especially when dealing with polygons defined by coordinates on a Cartesian plane, you can use the methods of coordinate geometry to find the lengths and slopes of sides, and subsequently, calculate the angles. This involves using trigonometric functions (like arctan) to find the angles.

Practical Applications

Understanding exterior angles is crucial in many fields:

• Architecture and Construction: Exterior angles are essential in designing structures, calculating angles for roof pitches, and ensuring structural stability.

• Engineering: Exterior angles are critical in designing bridges, roads, and other infrastructure projects.

• Cartography: The concept is applied in mapmaking and surveying to determine accurate angles and distances.

• Computer Graphics and Game Development: The understanding of angles and geometric calculations, including exterior angles, is essential for creating and rendering shapes accurately in computer graphics and video games.

• Navigation: Exterior angles play a crucial role in navigation systems, assisting in determining directions and paths.

Conclusion

Finding the measure of each exterior angle might seem complex initially, but by understanding the fundamental relationships between interior and exterior angles and applying the appropriate formulas and methods, you can effectively tackle any problem. Remember to always consider the type of polygon and the given information. Mastering these techniques will improve your problem-solving skills in geometry and broaden your understanding of spatial reasoning, enabling you to confidently tackle geometric challenges in various applications. Remember to practice regularly to solidify your understanding and develop proficiency in calculating exterior angles. This guide provides a strong foundation, but continued exploration and application of these concepts will undoubtedly enhance your mathematical abilities.