Find Area of Combined Rectangles - Complete Guide for 3rd Grade

Master finding area of combined rectangles: visual representations that make area calculation easier through partitioning shapes into smaller parts.

Complete Summary: Find Area of Combined Rectangles
Key Concepts & Definitions
Area
The amount of space inside a 2D shape. For rectangles, area is calculated by multiplying length by width (Area = length × width).
Combined Rectangles
Two or more rectangles joined together to form a larger shape. To find the total area, calculate each rectangle's area separately and add them together.
Decomposition
Breaking apart complex shapes into smaller, easier-to-work-with rectangles to simplify area calculation. This is the foundation of finding area of combined rectangles.
Total Area
The sum of all individual rectangle areas in a combined shape. Total Area = Area of Rectangle 1 + Area of Rectangle 2 + ...
\(A_{total} = A_1 + A_2 + A_3 + ... + A_n\)
Total Area Formula
Formula
\(Area = Length \times Width\)
Calculate area of a single rectangle
Combined Area
\(A_{total} = A_1 + A_2\)
Sum of individual rectangle areas
Decomposition
\(Total = R_1 + R_2 + R_3\)
Break shape into separate rectangles
Core Definitions:

Area: The measure of the space inside a 2D shape, measured in square units.

Combined Rectangles: Two or more rectangles joined together to form a larger shape. These shapes can be arranged side by side or stacked on top of each other.

Decomposition: The process of breaking a complex shape into smaller, simpler shapes to make area calculation easier.

Step-by-Step Method:
  1. Identify the shape: Recognize that you have a combined rectangle made of multiple smaller rectangles
  2. Separate the rectangles: Visually or physically separate each rectangle in the combined shape
  3. Measure dimensions: Find the length and width of each individual rectangle
  4. Calculate individual areas: Use Area = Length × Width for each rectangle
  5. Add the areas: Sum all individual areas to get the total area
Tip 1: Always label each rectangle with its dimensions before calculating.
Tip 2: Draw lines to separate rectangles clearly when working with complex shapes.
Tip 3: Check your work by counting unit squares when possible.
Tip 4: Remember to include units in your final answer (square units).
Common errors: Forgetting to add all rectangle areas, miscalculating dimensions, mixing up length and width.
Memory aid: "Add all the little areas to get the big area!"
Essential Rules:

• Individual Area: Calculate Area = Length × Width for each rectangle

• Total Area: Sum of all individual rectangle areas

• Decomposition: Break complex shapes into simpler rectangles

Solution: Exercises 1 to 3
1 Basic Combined Rectangles
Exercise 1
Find the total area of a combined rectangle made of two rectangles: one with dimensions 4 units by 3 units and another with dimensions 4 units by 2 units.
Concept:

Combined Rectangles: Two rectangles joined together, each with its own dimensions. Total area is the sum of individual areas.

Solution method:
  1. Identify the dimensions of each rectangle
  2. Calculate the area of each rectangle separately
  3. Add the areas together to get the total area
Rectangle 1
4 × 3 = 12
Rectangle 2
4 × 2 = 8
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Rectangle 1
\(4 \times 3 = 12\)
Rectangle 2
\(4 \times 2 = 8\)
Total
\(12 + 8 = 20\)
Step 1: Identify the dimensions

Rectangle 1: 4 units by 3 units

Rectangle 2: 4 units by 2 units

Step 2: Calculate area of Rectangle 1

Area = Length × Width = 4 × 3 = 12 square units

Step 3: Calculate area of Rectangle 2

Area = Length × Width = 4 × 2 = 8 square units

Step 4: Add the areas together

Total Area = 12 + 8 = 20 square units

\(Total\ Area = (4 \times 3) + (4 \times 2) = 12 + 8 = 20\)
Final answer:

The total area of the combined rectangles is 20 square units

Applied rules:

• Individual Area Calculation: Area = Length × Width

• Total Area Addition: Sum of all individual areas

• Decomposition: Break complex shape into simpler parts

2 Medium Difficulty
Exercise 2
A playground is made of two rectangular sections. One section is 6 feet by 4 feet and the other is 3 feet by 4 feet. What is the total area of the playground?
Concept:

Real-world Application: Combined rectangles can represent real spaces like playgrounds, gardens, or rooms.

Solution method:
  1. Identify the dimensions of each rectangular section
  2. Calculate the area of each section separately
  3. Add the areas to get the total playground area
Playground Section 1
6 × 4 = 24
Playground Section 2
3 × 4 = 12
Section 1
\(6 \times 4 = 24\)
Section 2
\(3 \times 4 = 12\)
Total
\(24 + 12 = 36\)
Step 1: Identify the dimensions

Section 1: 6 feet by 4 feet

Section 2: 3 feet by 4 feet

Step 2: Calculate area of Section 1

Area = Length × Width = 6 × 4 = 24 square feet

Step 3: Calculate area of Section 2

Area = Length × Width = 3 × 4 = 12 square feet

Step 4: Add the areas together

Total Area = 24 + 12 = 36 square feet

\(Total\ Area = (6 \times 4) + (3 \times 4) = 24 + 12 = 36\)
Final answer:

The total area of the playground is 36 square feet

Applied rules:

• Individual Area Calculation: Area = Length × Width

• Total Area Addition: Sum of all individual areas

• Real-world Context: Apply math to practical situations

3 Advanced Application
Exercise 3
A garden has two rectangular sections. Section A is 5 meters by 7 meters and Section B is 5 meters by 3 meters. Find the total area of the garden.
Concept:

Garden Planning: Combined rectangles can represent different areas of a garden for planning purposes.

Solution method:
  1. Identify the dimensions of each garden section
  2. Calculate the area of each section separately
  3. Add the areas to get the total garden area
Garden Section A
5 × 7 = 35
Garden Section B
5 × 3 = 15
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Section A
\(5 \times 7 = 35\)
Section B
\(5 \times 3 = 15\)
Total
\(35 + 15 = 50\)
Step 1: Identify the dimensions

Section A: 5 meters by 7 meters

Section B: 5 meters by 3 meters

Step 2: Calculate area of Section A

Area = Length × Width = 5 × 7 = 35 square meters

Step 3: Calculate area of Section B

Area = Length × Width = 5 × 3 = 15 square meters

Step 4: Add the areas together

Total Area = 35 + 15 = 50 square meters

\(Total\ Area = (5 \times 7) + (5 \times 3) = 35 + 15 = 50\)
Final answer:

The total area of the garden is 50 square meters

Applied rules:

• Individual Area Calculation: Area = Length × Width

• Total Area Addition: Sum of all individual areas

• Real-world Context: Apply math to practical situations

Solution: Exercises 4 to 5
4 Complex Combined Shape
Exercise 4
A room has an L-shaped floor plan made of two rectangles. Rectangle 1 is 8 feet by 5 feet, and Rectangle 2 is 3 feet by 5 feet. What is the total area of the floor?
Concept:

L-Shaped Floor Plan: Complex shapes can be decomposed into simpler rectangles to calculate total area.

Solution method:
  1. Visualize the L-shape as two separate rectangles
  2. Identify the dimensions of each rectangle
  3. Calculate the area of each rectangle
  4. Add the areas to get the total floor area
Rectangle 1
8 × 5 = 40
Rectangle 2
3 × 5 = 15
Rectangle 1
\(8 \times 5 = 40\)
Rectangle 2
\(3 \times 5 = 15\)
Total
\(40 + 15 = 55\)
Step 1: Visualize the L-shape

Break the L-shape into two rectangles: one 8×5 and one 3×5

Step 2: Calculate area of Rectangle 1

Area = Length × Width = 8 × 5 = 40 square feet

Step 3: Calculate area of Rectangle 2

Area = Length × Width = 3 × 5 = 15 square feet

Step 4: Add the areas together

Total Area = 40 + 15 = 55 square feet

\(Total\ Area = (8 \times 5) + (3 \times 5) = 40 + 15 = 55\)
Final answer:

The total area of the floor is 55 square feet

Applied rules:

• Shape Decomposition: Break complex shapes into simpler rectangles

• Individual Area Calculation: Area = Length × Width

• Total Area Addition: Sum of all individual areas

5 Multi-Rectangle Problem
Exercise 5
A building floor plan consists of three rectangular sections: Section X (4m × 6m), Section Y (4m × 2m), and Section Z (4m × 3m). Find the total area of the floor plan.
Concept:

Multi-Rectangle Floor Plan: More complex shapes with multiple rectangular sections require calculating each area separately.

Solution method:
  1. Identify the dimensions of each rectangular section
  2. Calculate the area of each section separately
  3. Add all areas to get the total floor area
Section X
4 × 6 = 24
Section Y
4 × 2 = 8
Section Z
4 × 3 = 12
Section X
\(4 \times 6 = 24\)
Section Y
\(4 \times 2 = 8\)
Section Z
\(4 \times 3 = 12\)
Total
\(24 + 8 + 12 = 44\)
Step 1: Identify all sections and dimensions

Section X: 4m × 6m, Section Y: 4m × 2m, Section Z: 4m × 3m

Step 2: Calculate area of Section X

Area = Length × Width = 4 × 6 = 24 square meters

Step 3: Calculate area of Section Y

Area = Length × Width = 4 × 2 = 8 square meters

Step 4: Calculate area of Section Z

Area = Length × Width = 4 × 3 = 12 square meters

Step 5: Add all areas together

Total Area = 24 + 8 + 12 = 44 square meters

\(Total\ Area = (4 \times 6) + (4 \times 2) + (4 \times 3) = 24 + 8 + 12 = 44\)
Final answer:

The total area of the floor plan is 44 square meters

Applied rules:

• Multiple Area Calculation: Calculate each rectangle's area separately

• Total Area Addition: Sum all individual areas

• Systematic Approach: Organize calculations to avoid missing sections

Questions & Answers

Question: What if the rectangles overlap? Do I still add their areas together?

Answer: Good question! For this lesson, we focus on rectangles that are next to each other without overlapping. If rectangles do overlap, you would subtract the overlapping area from the total. But in our combined rectangle problems, the rectangles are placed side by side or on top of each other without overlapping, so you always add their areas together.

Question: How do I know if a shape is made of combined rectangles or if it's just one big rectangle?

Answer: Look for distinct rectangular sections within the shape. Combined rectangles often have different dimensions in different parts of the shape. For example, an L-shape has two different rectangular sections. If the entire shape has consistent length and width measurements throughout, it's likely one big rectangle. If you see different measurements in different parts, it's probably made of combined rectangles.

Question: What if I count the squares instead of multiplying? Is that okay?

Answer: Counting squares is a great way to check your answer! It's especially helpful when you're first learning. But for larger rectangles, multiplying length by width is much faster and more efficient. Use counting to verify your multiplication answer, but aim to use multiplication as your primary method since it's quicker for bigger numbers.