Lesson 5: Angle Sums and Properties of Polygons

👨🏽‍🏫 Introduction

Have you ever wondered why windows, signs, or tiles come in specific shapes? Whether it’s the octagonal shape of a stop sign 🚦 or a tiled floor pattern ⬛, understanding how the angles in a polygon work is key to designing and analyzing structures!

In this lesson, we’ll focus on the angle properties of polygons — both regular and irregular — and learn how to calculate:

• 🧠 The total sum of interior angles
• 📐 The measure of each interior angle in regular polygons
• 🔄 The exterior angles and their consistent 360° property

🧠 Key Concepts

📌 1. What Is a Polygon?

• A polygon is a closed 2D shape with straight sides
• Regular polygons have all sides and all angles equal
• Irregular polygons have unequal sides and/or angles

📌 2. Sum of Interior Angles

The total interior angle sum of an n-sided polygon is given by:

📌 3. Each Interior Angle of a Regular Polygon

If all angles are equal (regular polygon), then:

📌 4. Exterior Angles

• Exterior angle = supplementary to the interior angle
• Sum of all exterior angles in any polygon =
• In a regular polygon:

🔍 Worked Examples

1. 1. Example 1: Find the sum of interior angles in a 9-sided polygon.
      
      🟢 Answer: 1260°

1. 1. Example 2: Find each interior angle of a regular heptagon (7 sides).
      
      🟢 Answer: ≈ 128.57°

1. 1. Example 3: Find one exterior angle of a regular dodecagon (12 sides).
      
      🟢 Answer: 30°

1. Example 4: If one exterior angle is 20°, how many sides does the regular polygon have?
   
   🟢 Answer: 18-sided polygon

✍🏽 Practice Exercises

🅰️ Section A – Interior Angles

1. Find the interior angle sum of a decagon (10 sides).
2. Find each interior angle of a regular pentagon.

🅱️ Section B – Exterior Angles

1. What is one exterior angle of a regular octagon?
2. A regular polygon has an exterior angle of 45°. How many sides does it have?

🅾️ Section C – Mixed

1. How many sides does a regular polygon have if each interior angle is 150°?
2. Find both the interior and exterior angles of a regular hexagon.

✅ Answers and Explanations

• A1:
• A2:
• B1:
• B2:
• C1:
• C2: Interior: , Exterior:

📌 Recap

• 🔺 Polygons are classified by the number of sides (n)
• 📐 Interior angle sum =
• 🔄 Exterior angles always add up to 360°
• ✅ In regular polygons, all angles and sides are equal

💭 Reflection Prompt

Take a look at your classroom windows, signs, or tiled floors. Can you tell how many sides those shapes have? What are their interior or exterior angles? Try calculating them!