External Angles Of A Polygon

Understanding External Angles of a Polygon: A Comprehensive Guide

External angles, also known as exterior angles, are a fundamental concept in geometry, particularly when dealing with polygons. This comprehensive guide will explore the properties of external angles, providing a detailed explanation suitable for students and anyone interested in deepening their understanding of geometric principles. We will cover various polygon types, explore their relationship with internal angles, and delve into practical applications. By the end, you'll have a solid grasp of external angles and their significance in geometry.

Introduction to Polygons and their Angles

Before diving into external angles, let's refresh our understanding of polygons. A polygon is a closed two-dimensional figure formed by joining three or more straight line segments. These segments are called the sides of the polygon, and the points where the sides meet are called vertices. Polygons are classified based on the number of sides they have: triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), and so on.

Each interior angle of a polygon is formed by two adjacent sides meeting at a vertex. The sum of the interior angles of a polygon depends on the number of sides. For example, the sum of the interior angles of a triangle is 180°, a quadrilateral is 360°, and so on. The formula for calculating the sum of interior angles (S) of an n-sided polygon is:

S = (n - 2) × 180°

Defining External Angles

An external angle of a polygon is formed by extending one side of the polygon and the adjacent side. Specifically, at each vertex, an external angle is formed by extending one of the sides. It's the angle between the extended side and the adjacent side. Notice that at each vertex, there are two external angles – one on each side. However, when we discuss external angles of a polygon, we typically refer to the consecutive external angles, one at each vertex. These angles lie outside the polygon.

Properties of External Angles

External angles possess several crucial properties:

• The sum of consecutive external angles of any polygon is always 360°. This is true regardless of the number of sides the polygon has. This is a remarkably consistent property, unlike the sum of interior angles which varies with the number of sides.

• Each external angle is supplementary to its corresponding interior angle. This means that the sum of an external angle and its adjacent interior angle is always 180°.

• The relationship between external and interior angles helps in finding unknown angles. If you know one, you can easily calculate the other using the supplementary angle property.

• Regular polygons have equal external angles. In a regular polygon (where all sides and angles are equal), all external angles are also equal. This simplifies calculations considerably.

Calculating External Angles

Let's look at how to calculate external angles in various scenarios:

1. Regular Polygons:

Calculating external angles for regular polygons is straightforward. Since the sum of external angles is always 360°, and all external angles are equal in a regular polygon, you can find the measure of a single external angle by dividing 360° by the number of sides (n):

External angle = 360° / n

For example, a regular hexagon (n=6) has an external angle of 360°/6 = 60°.

2. Irregular Polygons:

Calculating external angles for irregular polygons requires a slightly different approach. You'll need to know either the interior angles or other external angles to deduce the rest. Remember to use the supplementary angle relationship (internal angle + external angle = 180°) to find unknowns.

3. Using the Sum of Interior Angles:

You can also utilize the formula for the sum of interior angles to indirectly find external angles. First, calculate the sum of interior angles using the formula (n-2) x 180°. Then, find each individual interior angle if possible (by dividing the sum by the number of sides for regular polygons). Finally, use the supplementary angle property to determine each external angle.

Examples and Worked Problems

Let's work through a few examples to solidify our understanding:

Example 1: Finding the external angle of a regular pentagon.

A regular pentagon has 5 sides (n=5). The external angle is:

External angle = 360° / 5 = 72°

Example 2: Finding an unknown external angle in an irregular quadrilateral.

Suppose we have an irregular quadrilateral with three known external angles: 70°, 85°, and 100°. To find the fourth external angle, we use the fact that the sum of external angles is 360°:

360° = 70° + 85° + 100° + x x = 360° - 255° = 105°

Therefore, the fourth external angle is 105°.

Example 3: Using Interior Angles to Find External Angles.

Consider a triangle with interior angles of 60°, 70°, and 50°. The external angles are:

• External angle 1: 180° - 60° = 120°
• External angle 2: 180° - 70° = 110°
• External angle 3: 180° - 50° = 130°

Notice that 120° + 110° + 130° = 360°, confirming the property of the sum of external angles.

Explanation of the 360° Rule: A Deeper Dive

Why is the sum of external angles always 360°? This can be visualized by imagining "walking" around the perimeter of the polygon. At each vertex, you turn through an angle equal to the external angle. After completing a full circuit around the polygon, you've essentially made a complete 360° rotation. This intuitive explanation highlights the geometrical significance of the 360° rule.

Applications of External Angles

The concept of external angles isn't just a theoretical exercise; it finds practical applications in various fields:

• Architecture and Engineering: Understanding angles is crucial in designing stable structures. External angles help in calculating stress points and ensuring structural integrity.

• Cartography: External angles play a role in map-making, especially when dealing with the representation of curved surfaces on flat maps.

• Computer Graphics: In computer-aided design (CAD) and computer graphics, precise calculations of angles are essential for creating accurate and realistic images. External angles are used in various algorithms related to polygon manipulation.

• Robotics: In robotics, the precise movement of robotic arms and other mechanical components often relies on accurate angular calculations, including external angles.

Frequently Asked Questions (FAQ)

Q1: What is the difference between an interior angle and an exterior angle?

A1: An interior angle is formed inside the polygon by two adjacent sides meeting at a vertex. An exterior angle is formed outside the polygon by extending one side and considering the angle formed with the adjacent side.

Q2: Can an external angle be greater than 180°?

A2: No, an external angle, as defined, is always less than 180°. The angle formed by extending a line segment beyond the vertex will always be less than 180°. However, the reflex angle (the larger angle formed outside the polygon) can be greater than 180°.

Q3: What happens to the external angles of a polygon if its sides are extended in the opposite direction?

A3: Extending sides in the opposite direction will create a set of external angles supplementary to the initial set. The sum of these new external angles will also be 360°.

Q4: Are external angles relevant for concave polygons?

A4: Yes, the concept of external angles applies to concave polygons as well. However, you need to be careful when defining the external angle as it depends on the definition of the exterior region and which side you extend, and some may have reflex angles (greater than 180°). The rule about the sum of external angles still holds true.

Conclusion

External angles of polygons are a vital concept in geometry. Their consistent sum of 360° provides a powerful tool for solving geometric problems and understanding the properties of polygons. This property, combined with the supplementary relationship to interior angles, opens up various avenues for problem-solving, and finds its application across diverse fields. By grasping the fundamental properties and techniques presented in this guide, you are well-equipped to approach and tackle a wide range of geometric challenges involving external angles. Remember the key takeaway: the sum of the consecutive exterior angles of any polygon is always 360°, a fundamental truth in the world of geometry.