How Do You Find The Sum Of Exterior Angles

How Do You Find the Sum of Exterior Angles? A Comprehensive Guide

The sum of exterior angles of any polygon, no matter how many sides it has, is a remarkably consistent value. Understanding this concept is fundamental to geometry and has wide-ranging applications in various fields. This comprehensive guide will explore different methods to calculate this sum, providing you with a thorough understanding of the concept and its practical implications.

Understanding Exterior Angles

Before diving into the methods, let's establish a clear definition. An exterior angle is formed by extending one side of a polygon beyond a vertex. At each vertex, there are two exterior angles, but we typically consider only one—the one that forms a linear pair with the interior angle at that vertex. A linear pair of angles adds up to 180 degrees.

Key characteristics of exterior angles:

• Supplementary to Interior Angles: An exterior angle and its corresponding interior angle are supplementary, meaning their sum is 180°.
• One per Vertex: Only one exterior angle is considered at each vertex.
• Total Sum Consistency: The remarkable fact is that the sum of all exterior angles of any polygon is always the same.

Method 1: The Intuitive Approach - Walking Around the Polygon

Imagine you're walking around a polygon, always turning at each corner to follow the sides. At each corner, the amount you turn is equal to the exterior angle at that corner. When you complete your walk and return to your starting point, you've made a complete 360° turn. This illustrates that the sum of the exterior angles is 360°.

This method, while intuitive, offers a compelling visual understanding of why the sum remains constant regardless of the polygon's shape or the number of sides. It provides a strong foundation for understanding more formal methods.

Why this works: The process of walking around the polygon is equivalent to rotating a total of 360°. Each turn corresponds to an exterior angle. Therefore, the sum of exterior angles equals 360°.

Method 2: Using the Relationship with Interior Angles

This method leverages the relationship between interior and exterior angles. We know that the sum of the interior angles of a polygon with 'n' sides is given by the formula:

(n - 2) * 180°

Since each exterior angle and its corresponding interior angle are supplementary (add up to 180°), we can use this knowledge to find the sum of exterior angles.

Let's consider a triangle (n=3). The sum of its interior angles is (3-2) * 180° = 180°. Each interior angle pairs with an exterior angle that sums to 180°. Therefore, the sum of exterior angles is 3 * 180° - 180° = 360°.

Now, let's generalize this for an n-sided polygon. The sum of the interior angles is (n - 2) * 180°. The sum of the interior and exterior angles together is n * 180° (since there are n vertices, each with a supplementary pair). Therefore, the sum of exterior angles is:

n * 180° - (n - 2) * 180° = 360°

This method demonstrates algebraically why the sum of exterior angles is always 360°. It's a more rigorous approach compared to the intuitive "walking around" method.

Method 3: The Decomposition Method (for Irregular Polygons)

For irregular polygons (polygons where sides and angles are not all equal), calculating interior angles individually can be tedious. The decomposition method provides an alternative. This involves breaking down the irregular polygon into several triangles by drawing diagonals from one vertex.

Steps:

1. Choose a Vertex: Select any vertex of the irregular polygon.
2. Draw Diagonals: Draw diagonals from the chosen vertex to all other non-adjacent vertices. This will divide the polygon into several triangles. The number of triangles will be (n-2), where 'n' is the number of sides.
3. Sum of Angles in Triangles: The sum of the interior angles of each triangle is 180°. The total sum of the angles in all the triangles will be (n-2) * 180°.
4. Relate to Exterior Angles: While this sum relates to the interior angles, we can still derive the exterior angle sum. Remember, each interior angle has a supplementary exterior angle. The sum of all the angles (interior + exterior) around the polygon is 360°.

This method is valuable because it visually demonstrates the concept and provides an alternative approach to calculating angles in complex polygons. While it doesn't directly calculate the exterior angle sum, it shows how it links to the total angular sum of the polygon.

Applications of the Sum of Exterior Angles

Understanding the sum of exterior angles isn't just a theoretical exercise. It has practical applications in:

• Cartography: Determining the total angular deviation in map projections.
• Engineering: Designing structures, ensuring angles align correctly.
• Computer Graphics: Creating polygon-based models and animations.
• Robotics: Programming robotic movements and orientations.
• Navigation: Calculating headings and course corrections.

Solving Problems Related to Exterior Angles

Let's work through some example problems:

Problem 1: Find the measure of each exterior angle of a regular pentagon.

Solution: A regular pentagon has 5 sides. The sum of its exterior angles is always 360°. Since it's regular, each exterior angle is equal. Therefore, each exterior angle measures 360°/5 = 72°.

Problem 2: A polygon has an exterior angle sum of 360°. How many sides does it have?

Solution: The sum of exterior angles is always 360°, regardless of the number of sides. This question is a bit of a trick question to ensure understanding of the fundamental principle. Therefore, the polygon can have any number of sides (greater than 2).

Problem 3: An irregular hexagon has exterior angles of 70°, 60°, x, 80°, 50°, and 90°. Find the value of x.

Solution: The sum of the exterior angles of any hexagon is 360°. Therefore: 70° + 60° + x + 80° + 50° + 90° = 360° Solving for x, we get x = 10°.

Advanced Considerations

The concept extends beyond simple polygons. It applies to:

• Non-convex polygons: Polygons with at least one interior angle greater than 180°. The exterior angle calculation still holds, ensuring that external angles on the concave side are taken as greater than 180°.
• Three-dimensional shapes: The concept can be extended to analyze the exterior angles of polyhedra. Although slightly more complex, the underlying principles remain consistent.

Conclusion

The sum of exterior angles of any polygon is invariably 360°. This seemingly simple geometric concept offers a profound understanding of polygon properties and finds diverse applications across various fields. Through different approaches—the intuitive 'walking around' method, the algebraic approach using interior angles, and the decomposition method—we've explored various ways to grasp this essential geometric principle. Mastering this concept forms a strong foundation for further exploration in geometry and related disciplines.