Lesson 4: Special Triangles and Polygons

👨🏽‍🏫 Introduction

Why are triangle road signs almost always equilateral? And why do engineers love hexagons when building strong structures? From traffic design to honeycombs, special triangles and polygons appear everywhere in the world around us. 🚧🍯

In this lesson, you’ll learn to recognize and work with common special triangles (like equilateral and right-angled) and regular polygons (shapes with equal sides and angles). You’ll also learn how to calculate angles inside these shapes using simple rules!

• 🔺 Identify special triangle types and properties
• ⬡ Explore common regular polygons (triangle, square, pentagon, hexagon…)
• 📏 Calculate interior and exterior angles

🧠 Key Concepts

📌 1. Special Triangles

• Equilateral triangle: All sides and angles are equal (each angle is )
• Isosceles triangle: Two equal sides and two equal angles
• Right-angled triangle: One angle is

📐 2. Regular Polygons

A regular polygon has all sides and all angles equal. Examples include:

• Equilateral triangle (3 sides)
• Square (4 sides)
• Pentagon (5 sides)
• Hexagon (6 sides)
• Octagon (8 sides)

📏 3. Interior and Exterior Angles

• Sum of interior angles of any polygon = , where n is the number of sides
• Interior angle of a regular polygon =
• Exterior angle of a regular polygon =

🔍 Worked Examples

1. 1. Example 1 – Interior Angles in a Polygon:
      Find the sum of interior angles of a hexagon (6 sides).
      
      🟢 Answer: 720°

1. 1. Example 2 – Interior Angle of a Regular Pentagon:
      
      🟢 Each interior angle is 108°

1. 1. Example 3 – Exterior Angle of a Regular Octagon:
      
      🟢 Each exterior angle = 45°

1. Example 4 – Angle in an Equilateral Triangle:
   
   🟢 Each angle = 60°

✍🏽 Practice Exercises

🅰️ Section A – Special Triangles

1. What are the angles in an equilateral triangle?
2. In an isosceles triangle, if the base angle is 70°, what is the vertex angle?

🅱️ Section B – Regular Polygons

1. Find the sum of interior angles in a regular octagon.
2. Find one interior angle in a regular hexagon.

🅾️ Section C – Exterior Angles

1. Find the exterior angle of a regular decagon (10 sides).
2. If one exterior angle of a polygon is 30°, how many sides does the polygon have?

✅ Answers and Explanations

• A1: 60°, 60°, 60° (all equal)
• A2:
• B1:
• B2:
• C1:
• C2:

📌 Recap

• 🔺 Equilateral triangles have 3 equal sides and 60° angles
• 🟥 Isosceles triangles have 2 equal sides and 2 equal angles
• ⬡ Regular polygons have equal sides and angles
• 📏 Interior angle sum of an n-sided polygon is
• 🔄 Exterior angle of a regular polygon is

💭 Reflection Prompt

Think of a stop sign (octagon) or tiled bathroom floor (hexagons). Why might builders choose these shapes? Can you identify a polygon in your surroundings and calculate one of its angles?