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Lesson 4-3

Complex Shape Areas

Calculate areas of composite shapes by breaking them into simple parts. Master area calculation through garden design projects and real-world applications.

Learning Scenario: Garden Design Project

Scenario: You're designing a garden! The garden has a rectangular lawn (5m × 3m), a square flower bed (2m × 2m), and a triangular herb garden (base 3m, height 2m). You need to calculate the total area to know how much grass seed and soil to buy. How do you find the total area of this complex shape?

This project will teach you the "decomposition method" - breaking complex shapes into simple parts, calculating each area, then adding them together!

Decomposition Method

Break complex shapes into simple shapes (rectangles, squares, triangles) that you know how to calculate.

Steps:

  1. 1. Identify simple shapes
  2. 2. Calculate each area separately
  3. 3. Add all areas together
  4. 4. Check your work
Area Formulas

Use these basic formulas for the simple shapes you'll encounter.

Formulas:

  • • Rectangle: A = length × width
  • • Square: A = side × side
  • • Triangle: A = ½ × base × height
  • • Circle: A = π × radius²
Step-by-Step Learning
1

Identify Simple Shapes

Look at the complex shape and identify the simple shapes it's made of. Draw lines to separate them if needed.

Example: L-shaped room = Rectangle + Square

2

Calculate Each Area

Use the appropriate formula for each simple shape. Make sure you have the correct measurements.

Example: Rectangle: 5m × 3m = 15m², Square: 2m × 2m = 4m²

3

Add Areas Together

Add all the individual areas to get the total area of the complex shape.

Example: Total = 15m² + 4m² = 19m²

Interactive Activities

Activity 1: Shape Decomposition

Break these complex shapes into simple parts:

  • • L-shaped room (8m × 6m with 4m × 4m cutout)
  • • T-shaped garden (10m × 3m + 3m × 4m)
  • • U-shaped patio (6m × 4m with 2m × 2m cutout)
  • • Plus-shaped floor plan

Activity 2: Area Calculation

Calculate the total area:

  • • Rectangle: 6m × 4m
  • • Square: 3m × 3m
  • • Triangle: base 4m, height 3m
  • • Total area = ?
Practice Problems

Problem Set 1: Basic Decomposition

1. L-shape: Rectangle 8m×6m + Square 3m×3m. Total area?

2. T-shape: Rectangle 10m×4m + Rectangle 2m×6m. Total area?

3. U-shape: Rectangle 6m×5m - Square 2m×2m. Total area?

4. Plus-shape: Rectangle 8m×3m + Rectangle 3m×6m. Total area?

Problem Set 2: Mixed Shapes

5. Rectangle 5m×4m + Triangle base 3m, height 2m. Total area?

6. Square 6m×6m + Rectangle 2m×4m. Total area?

7. Rectangle 7m×5m - Triangle base 2m, height 3m. Total area?

8. Two rectangles: 4m×3m + 3m×2m. Total area?

Problem Set 3: Real-World Applications

9. Garden: Lawn 8m×6m + Flower bed 2m×2m. Total area?

10. Room: Main area 5m×4m + Closet 1m×2m. Total area?

11. Patio: Rectangle 6m×4m - Pool 2m×2m. Total area?

12. Floor: Living room 8m×6m + Kitchen 4m×3m. Total area?

Common Mistakes to Avoid

❌ Forgetting to subtract cutout areas

Wrong: Adding all areas when some should be subtracted

Correct: Subtract cutout areas from the main shape

❌ Using wrong measurements

Wrong: Using the wrong length or width for a shape

Correct: Double-check measurements for each simple shape

Key Takeaways

Break complex shapes into simple shapes you know

Calculate each area separately using the right formula

Add areas together (subtract for cutouts)

Always double-check your measurements and calculations