Polygon Interior Exterior Angles Worksheet

Mastering Polygons: A practical guide to Interior and Exterior Angles with Worksheets

Understanding interior and exterior angles of polygons is a fundamental concept in geometry. Plus, this practical guide provides a step-by-step approach to mastering this topic, complete with practice worksheets and explanations to solidify your understanding. So naturally, whether you're a student tackling geometry for the first time or looking for a refresher, this resource will equip you with the knowledge and tools to confidently solve problems involving polygon angles. We'll explore the formulas, the reasoning behind them, and provide ample opportunities to practice.

Introduction to Polygons and Their Angles

A polygon is a closed two-dimensional shape formed by connecting 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.

Every polygon has both interior and exterior angles.

• Interior Angles: These are the angles formed inside the polygon at each vertex. The sum of the interior angles of a polygon depends on the number of sides it has.

• Exterior Angles: These are the angles formed outside the polygon by extending one of its sides. At each vertex, there's one interior angle and one exterior angle that are supplementary, meaning they add up to 180 degrees.

Understanding the relationship between the number of sides and the sum of interior and exterior angles is crucial for solving various geometry problems.

Finding the Sum of Interior Angles

The sum of the interior angles of a polygon is directly related to its number of sides (n). The formula for calculating this sum is:

(n - 2) * 180°

Where:

• n represents the number of sides of the polygon.

Example: Find the sum of the interior angles of a hexagon (6 sides).

Using the formula: (6 - 2) * 180° = 4 * 180° = 720°

The sum of the interior angles of a hexagon is 720°.

Finding the Measure of Each Interior Angle in a Regular Polygon

A regular polygon is a polygon where all sides are equal in length and all angles are equal in measure. To find the measure of each interior angle in a regular polygon, we can use the following formula:

[(n - 2) * 180°] / n

Where:

• n represents the number of sides of the polygon.

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

Using the formula: [(5 - 2) * 180°] / 5 = (3 * 180°) / 5 = 540° / 5 = 108°

Each interior angle of a regular pentagon measures 108°.

Finding the Sum of Exterior Angles

The sum of the exterior angles of any polygon (regardless of whether it's regular or irregular) is always 360°. This leads to this is a fundamental property of polygons. This fact is independent of the number of sides.

Finding the Measure of Each Exterior Angle in a Regular Polygon

In a regular polygon, all exterior angles are equal in measure. To find the measure of each exterior angle, we can use the following formula:

360° / n

Where:

• n represents the number of sides of the polygon.

Example: Find the measure of each exterior angle of a regular octagon (8 sides).

Using the formula: 360° / 8 = 45°

Each exterior angle of a regular octagon measures 45°.

Relationship Between Interior and Exterior Angles

Remember, each interior angle and its corresponding exterior angle are supplementary; they add up to 180°. This relationship can be used to solve problems where you know one angle and need to find the other.

Worksheet 1: Finding the Sum of Interior Angles

Instructions: Find the sum of the interior angles for each polygon.

1. Triangle (3 sides)
2. Quadrilateral (4 sides)
3. Heptagon (7 sides)
4. Decagon (10 sides)
5. Dodecagon (12 sides)

Worksheet 2: Finding the Measure of Each Interior Angle in a Regular Polygon

Instructions: Find the measure of each interior angle for each regular polygon.

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1. Square (4 sides)
2. Pentagon (5 sides)
3. Hexagon (6 sides)
4. Nonagon (9 sides)
5. 15-sided polygon

Worksheet 3: Finding the Measure of Each Exterior Angle in a Regular Polygon

Instructions: Find the measure of each exterior angle for each regular polygon.

1. Triangle (3 sides)
2. Quadrilateral (4 sides)
3. Octagon (8 sides)
4. Pentagon (5 sides)
5. Heptagon (7 sides)

Worksheet 4: Mixed Problems

Instructions: Solve the following problems.

1. A regular polygon has an interior angle of 150°. How many sides does it have?
2. The sum of the interior angles of a polygon is 1800°. How many sides does it have?
3. Each exterior angle of a regular polygon measures 20°. How many sides does it have?
4. Find the measure of each exterior angle of a regular hexagon. Then, find the measure of each interior angle.
5. A polygon has 11 sides. What is the sum of its interior angles?

Explanation of Solutions (Worksheet 4)

1. Solution: If the interior angle is 150°, the exterior angle is 180° - 150° = 30°. Since the sum of exterior angles is 360°, the number of sides is 360° / 30° = 12 sides.

2. Solution: Use the formula (n - 2) * 180° = 1800°. Solve for n: n - 2 = 10, n = 12 sides.

3. Solution: Since the sum of exterior angles is 360°, the number of sides is 360° / 20° = 18 sides.

4. Solution: Each exterior angle of a regular hexagon is 360° / 6 = 60°. Each interior angle is 180° - 60° = 120°.

5. Solution: The sum of interior angles of an 11-sided polygon is (11 - 2) * 180° = 1620°.

Frequently Asked Questions (FAQs)

• Q: What is the difference between a regular and irregular polygon?

  • A: A regular polygon has all sides equal in length and all angles equal in measure. An irregular polygon does not have this property.

• Q: Can the interior angle of a polygon be greater than 180°?

  • A: No, the interior angle of a convex polygon (a polygon where all interior angles are less than 180°) cannot be greater than 180°. Still, concave polygons (polygons with at least one interior angle greater than 180°) are possible.

• Q: Why is the sum of exterior angles always 360°?

  • A: Imagine walking around the perimeter of a polygon. At each vertex, you turn through an exterior angle. By the time you return to your starting point, you've made a complete 360° turn.

• Q: Can I use these formulas for non-convex polygons?

  • A: The formula for the sum of interior angles, (n-2) * 180°, holds true for both convex and concave polygons. Still, the individual interior angles of a concave polygon will have at least one angle greater than 180°. The sum of exterior angles remains 360° even for concave polygons, although the individual exterior angles may be negative.

• Q: What are some real-world applications of polygon angle calculations?

  • A: Polygon angle calculations are used in many fields, including architecture (designing buildings and structures), engineering (calculating forces and stresses), computer graphics (creating and manipulating shapes), and cartography (creating maps).

Conclusion

Mastering the concepts of interior and exterior angles of polygons is essential for success in geometry and related fields. And by understanding the formulas and practicing with the provided worksheets, you'll develop a strong foundation in this important area of mathematics. Consistent practice is key to building confidence and proficiency in solving polygon angle problems. In practice, remember to put to use the relationships between interior and exterior angles to solve a variety of problems. Through understanding and application, you'll move beyond simply memorizing formulas to truly grasping the underlying principles.