How Do You Find The Sum Of Exterior Angles

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

Understanding the sum of exterior angles of polygons is a fundamental concept in geometry. This comprehensive guide will explore this topic in detail, providing you with a clear understanding of the principles involved, various methods for calculating the sum, and practical applications. We'll delve into different types of polygons, including triangles, quadrilaterals, and n-sided polygons, ensuring you grasp the concept fully. By the end of this article, you'll be confident in calculating the sum of exterior angles for any polygon.

Understanding Exterior Angles

Before we jump into calculations, let's define what an exterior angle is. An exterior angle is formed by extending one side of a polygon. 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 formed by extending one side and another formed by extending the other side. However, we usually consider only one exterior angle at each vertex when discussing the sum of exterior angles. We typically select the exterior angle that is supplementary to the interior angle. This means that the interior angle and the chosen exterior angle add up to 180 degrees.

The Sum of Exterior Angles Theorem: A Cornerstone of Geometry

The most crucial fact about exterior angles is this: the sum of the exterior angles of any convex polygon, one at each vertex, is always 360 degrees. This holds true regardless of the number of sides the polygon has. This is a powerful theorem that simplifies many geometric problems. Let's explore why this is true.

Visualizing the Theorem

Imagine walking around the perimeter of a polygon, always turning at each vertex. At each turn, you're changing your direction. The amount you turn is the exterior angle at that vertex. When you complete a full circuit of the polygon and return to your starting point, you've made a full 360-degree turn. This visual representation intuitively explains why the sum of exterior angles is always 360 degrees.

Mathematical Proof (for Convex Polygons)

While the visual explanation is intuitive, a more rigorous mathematical proof can be constructed using the properties of interior angles. Consider a convex polygon with 'n' sides. The sum of its interior angles is given by the formula (n-2) * 180 degrees. Each interior angle and its corresponding exterior angle are supplementary (add up to 180 degrees). Let's represent the interior angles as I₁, I₂, ..., Iₙ and the exterior angles as E₁, E₂, ..., Eₙ. We know that:

• I₁ + E₁ = 180
• I₂ + E₂ = 180
• ...
• Iₙ + Eₙ = 180

Summing these equations, we get:

(I₁ + I₂ + ... + Iₙ) + (E₁ + E₂ + ... + Eₙ) = 180n

Since the sum of interior angles is (n-2) * 180, we can substitute:

(n-2) * 180 + (E₁ + E₂ + ... + Eₙ) = 180n

Simplifying the equation, we find:

(E₁ + E₂ + ... + Eₙ) = 180n - 180n + 360

Therefore:

(E₁ + E₂ + ... + Eₙ) = 360 degrees

This proves that the sum of exterior angles of any convex polygon is always 360 degrees.

Calculating the Sum of Exterior Angles: Methods and Examples

The simplest method for calculating the sum of exterior angles is to use the theorem directly: The sum is always 360 degrees. However, understanding how to calculate individual exterior angles is also crucial.

Method 1: Using the Interior Angles

If you know the interior angles of the polygon, you can easily find the exterior angles. Remember that an interior angle and its corresponding exterior angle are supplementary. Therefore:

Exterior Angle = 180 degrees - Interior Angle

Once you have all the exterior angles, simply add them together to obtain the sum (which should always be 360 degrees).

Example: A quadrilateral has interior angles of 70°, 100°, 110°, and 80°. Let's calculate the sum of its exterior angles:

• Exterior angle 1: 180° - 70° = 110°
• Exterior angle 2: 180° - 100° = 80°
• Exterior angle 3: 180° - 110° = 70°
• Exterior angle 4: 180° - 80° = 100°

Sum of exterior angles: 110° + 80° + 70° + 100° = 360°

Method 2: Using the Number of Sides (for Regular Polygons)

For regular polygons (polygons with all sides and angles equal), calculating the exterior angle is particularly straightforward. Since all exterior angles are equal, you can find one exterior angle and multiply by the number of sides. The formula for a regular polygon with 'n' sides is:

Exterior Angle = 360 degrees / n

Example: A regular hexagon (6 sides) has an exterior angle of 360°/6 = 60°. The sum of its exterior angles is 6 * 60° = 360°.

Applications of the Sum of Exterior Angles Theorem

The theorem about the sum of exterior angles has diverse applications in various fields, including:

• Cartography: Creating accurate maps involves understanding the angles of polygons representing geographical areas.
• Architecture and Engineering: Designing buildings and structures often requires precise calculations of angles, ensuring structural integrity and stability.
• Computer Graphics: Generating 3D models and animations relies heavily on geometric principles, including the calculation of angles in polygons.
• Game Development: Many video games utilize polygon-based rendering. Accurate angle calculations are crucial for realistic graphics.

Beyond Convex Polygons: Concave Polygons and the Sum of Exterior Angles

The theorem we've discussed applies specifically to convex polygons – polygons where all interior angles are less than 180 degrees. For concave polygons (polygons with at least one interior angle greater than 180 degrees), the concept of exterior angles needs careful consideration.

When dealing with concave polygons, the exterior angle at a reflex (concave) angle will be negative. While the sum of the magnitudes of the exterior angles might not be 360 degrees, the algebraic sum of the exterior angles (considering the reflex angles as negative) will still be 360 degrees.

Conclusion: Mastering the Sum of Exterior Angles

Understanding the sum of exterior angles is a fundamental skill in geometry with wide-ranging applications. The theorem that the sum is always 360 degrees for convex polygons provides a powerful tool for solving various geometric problems. This article has equipped you with the knowledge and methods to confidently calculate the sum of exterior angles for any convex polygon, and provided insight into the complexities of handling concave polygons. Remember to practice applying these methods to solidify your understanding. With consistent practice, you’ll become proficient in solving problems involving exterior angles and appreciate their significance in various fields.