Worksheet: Area of Composite Rectangular Shapes in Grade 4

Complete worksheet with detailed solutions and practice exercises for composite rectangular shapes.

Solution: Exercises 1 to 3
1 Rectangle Plus Rectangle
Exercise 1
Find the area of a figure made up of two rectangles: one 6 cm by 4 cm and another 3 cm by 4 cm placed side by side.
Definitions:

Composite Rectangular Shape: A shape made up of two or more rectangles

Area: The space inside a shape measured in square units

Basic Shapes: Rectangles that we can easily calculate

Calculation method:
  1. Separate the composite figure into basic rectangles
  2. Calculate the area of each rectangle
  3. Add all the areas together
Rectangle 1
6 × 4 = 24 cm²
Rectangle 2
3 × 4 = 12 cm²
Total Area
24 + 12 = 36 cm²
Step 1: Separate the figure

Rectangle 1: 6 cm by 4 cm

Rectangle 2: 3 cm by 4 cm

Step 2: Calculate area of each rectangle

Area of Rectangle 1 = 6 × 4 = 24 square cm

Area of Rectangle 2 = 3 × 4 = 12 square cm

Step 3: Add the areas together

Total Area = 24 + 12 = 36 square cm

Total Area = 36 cm²
Final answer:

The area of the composite rectangular shape is 36 square centimeters.

Applied rules:

• Separation Method: Break composite figures into basic rectangles

• Area Addition: Add individual areas to get total area

• Rectangle Area: Length × Width

2 Rectangle Minus Rectangle
Exercise 2
A large rectangle measures 8 cm by 6 cm. A smaller rectangle measuring 3 cm by 2 cm is cut out from one corner. Find the area of the remaining figure.
Definition:

Subtracted Area: When part of a shape is removed, subtract its area from the whole

Large Rectangle
8 × 6 = 48 cm²
Small Rectangle
3 × 2 = 6 cm²
Remaining Area
48 - 6 = 42 cm²
Step 1: Calculate area of large rectangle

Area = 8 × 6 = 48 square cm

Step 2: Calculate area of small rectangle

Area = 3 × 2 = 6 square cm

Step 3: Subtract the smaller area

Remaining Area = 48 - 6 = 42 square cm

Remaining Area = 42 cm²
Final answer:

The area of the remaining figure is 42 square centimeters.

Applied rules:

• Subtraction Method: When a part is removed, subtract its area

• Whole minus Part: Large area - Small area = Remaining area

• Rectangle Area: Length × Width

3 L-Shaped Figure
Exercise 3
An L-shaped figure is made by placing a 5 cm by 3 cm rectangle on top of a 7 cm by 2 cm rectangle, sharing a 3 cm side. Find the total area.
Definition:

L-Shaped Figure: A composite rectangular shape that looks like the letter L

Top Rectangle
5 × 3 = 15 cm²
Bottom Rectangle
7 × 2 = 14 cm²
Total Area
15 + 14 = 29 cm²
Step 1: Identify the rectangles

Top rectangle: 5 cm by 3 cm

Bottom rectangle: 7 cm by 2 cm

Step 2: Calculate area of each rectangle

Top rectangle: 5 × 3 = 15 square cm

Bottom rectangle: 7 × 2 = 14 square cm

Step 3: Add the areas together

Total Area = 15 + 14 = 29 square cm

Step 4: Verify the solution

Check that we haven't counted overlapping areas twice

Total Area = 29 cm²
Final answer:

The area of the L-shaped figure is 29 square centimeters.

Applied rules:

• Separation Method: Break L-shape into rectangles

• Area Addition: Add individual areas to get total

• No Overlap: Make sure not to count the same area twice

Formulas and Methods
Area = Sum of Areas of Rectangles
Composite Rectangular Shape Area
Area of Rectangle = Length × Width
Rectangle Area
Area of Square = Side²
Square Area
Key definitions:

Composite Rectangular Shape: A shape made up of two or more rectangles

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

Rectangles: Four-sided shapes with four right angles and opposite sides equal

Decomposition: Breaking a complex shape into simpler rectangular parts

Problem-solving method:
  1. Examine the figure: Look for recognizable rectangular shapes
  2. Decompose: Separate the figure into non-overlapping rectangles
  3. Calculate: Find the area of each rectangle
  4. Combine: Add or subtract areas depending on the configuration
  5. Verify: Check that your answer makes sense
Tip 1: Always draw lines to separate the rectangles clearly.
Tip 2: If parts are removed, subtract those areas.
Tip 3: Don't count overlapping areas twice.
Tip 4: Label each rectangle with its dimensions.
Common errors: Forgetting to add all parts, counting overlapping areas twice, misidentifying rectangles.
Memorization tip: Remember "Break it down, calculate each part, add them up".
Key Concepts and Properties
Total Area = A₁ + A₂ + ... + Aₙ
Area Addition Principle
Rectangle Area
A = L × W
Length × Width
Square Area
A = s²
Side × Side
Composite Area
A = Σ(Aᵢ)
Sum of all parts
Important properties:

Decomposition: Any composite rectangular shape can be broken down into individual rectangles

Additivity: The area of a whole equals the sum of the areas of its parts

Non-overlapping: Parts must not overlap when calculating total area

Measurement: Area is always expressed in square units

Composite rectangular shape strategy:
  1. Visualize: Look at the composite shape and identify individual rectangles
  2. Separate: Mentally or physically draw lines to separate shapes
  3. Measure: Identify necessary dimensions for each rectangle
  4. Calculate: Find the area of each rectangle
  5. Combine: Add or subtract areas as needed
Tip 1: Look for rectangles in the composite shape.

Tip 2: If a rectangle appears to be cut out, subtract its area.
Tip 3: Always include units in your final answer.
Essential principles to remember:

• Area Addition: Total area = Sum of all individual areas

• Decomposition: Break complex shapes into simple rectangles

• Rectangle Area: A = Length × Width

• Square Area: A = Side × Side

• Subtraction: When parts are removed, subtract their areas

Visual aid: Rectangles have length and width
Visual aid: Squares have all sides equal

Comprehensive Summary: Area of Composite Rectangular Shapes

Definitions of Key Concepts

Composite Rectangular Shape: A shape made up of two or more rectangles joined together.

Area: The amount of space inside a 2-dimensional shape, measured in square units (cm², m², etc.).

Rectangle: A four-sided shape with four right angles and opposite sides equal in length.

Decomposition: The process of breaking down a complex shape into simpler, recognizable rectangular parts.

Additivity Property: The area of a whole shape equals the sum of the areas of its non-overlapping parts.

Core Rules and Principles

Area Addition Rule: Total Area = Area of Rectangle 1 + Area of Rectangle 2 + ...

Area Subtraction Rule: When a part is removed, subtract its area from the whole: Total Area = Large Area - Small Area

Rectangle Area: Area = Length × Width

Square Area: Area = Side × Side = Side²

Non-Overlap Principle: When decomposing, ensure rectangles don't overlap to avoid double-counting.

Dimension Consistency: All measurements must use the same units.

Step-by-Step Methods

  1. Examine the Figure: Look carefully at the composite shape and identify recognizable rectangles within it.
  2. Draw Separation Lines: Mentally or physically draw lines to separate the composite figure into distinct rectangles.
  3. Label Dimensions: Identify and label the necessary dimensions for each rectangle.
  4. Calculate Individual Areas: Use the formula A = Length × Width to find the area of each rectangle.
  5. Combine Areas: Add all areas together if rectangles are joined, or subtract if parts are removed.
  6. Check Your Work: Verify that your answer makes sense and units are correct.

Examples: From Simple to Advanced

Simple Example: Two rectangles side by side - 4×3 and 2×3. Area = (4×3) + (2×3) = 12 + 6 = 18 cm²

Intermediate Example: L-shaped figure made of 5×2 and 3×4 rectangles. Area = (5×2) + (3×4) = 10 + 12 = 22 cm²

Advanced Example: A large rectangle (8×6) with a smaller rectangle (2×3) cut out. Area = (8×6) - (2×3) = 48 - 6 = 42 cm²

Tips, Tricks, and Common Pitfalls

Tips:

  • Always draw lines to clearly separate rectangles
  • Label each rectangle with its dimensions
  • Double-check that you haven't counted overlapping areas twice
  • Remember: when parts are removed, subtract their areas
  • Include units in your final answer

Common Pitfalls:

  • Forgetting to add all parts of the composite figure
  • Counting overlapping areas twice
  • Misidentifying rectangles or their dimensions
  • Using different units for different parts
  • Adding when you should subtract (or vice versa)

Key Notes for Memorization

Memory Aid: "Break it down, calculate each part, add them up!"

Pattern Recognition: Look for rectangles in complex shapes

Add vs Subtract: Add when rectangles are combined, subtract when parts are removed

Verification: Always check if your answer makes sense compared to the original figure

Unit Consistency: Square units for area, all measurements in the same unit

Practice Problems: Area of Composite Rectangular Shapes

Instructions:

Find the area of each composite rectangular shape. Show your work by breaking the shape into individual rectangles and calculating their areas.

1 T-Shaped Figure (5 points)

A T-shaped figure is formed by placing a 4 cm by 2 cm rectangle on top of a 6 cm by 3 cm rectangle, centered on top. Find the total area.

Solution:

Top rectangle: 4 × 2 = 8 cm²

Bottom rectangle: 6 × 3 = 18 cm²

Total Area = 8 + 18 = 26 cm²

2 Frame Shape (6 points)

A rectangular picture frame has outer dimensions of 10 cm by 8 cm and inner dimensions of 6 cm by 4 cm. Find the area of the frame itself.

Solution:

Outer rectangle: 10 × 8 = 80 cm²

Inner rectangle: 6 × 4 = 24 cm²

Frame area = 80 - 24 = 56 cm²

3 Cross Shape (7 points)

A cross is made of two rectangles: one horizontal rectangle 8 cm by 2 cm and one vertical rectangle 2 cm by 6 cm. They intersect in a 2 cm by 2 cm square. Find the total area.

Solution:

Horizontal rectangle: 8 × 2 = 16 cm²

Vertical rectangle: 2 × 6 = 12 cm²

Intersection: 2 × 2 = 4 cm²

Total Area = 16 + 12 - 4 = 24 cm²

4 L-Shape Variation (5 points)

An L-shaped figure has a base rectangle of 7 cm by 3 cm and an attached rectangle of 4 cm by 2 cm extending from the longer side. Find the total area.

Solution:

Base rectangle: 7 × 3 = 21 cm²

Attached rectangle: 4 × 2 = 8 cm²

Total Area = 21 + 8 = 29 cm²

5 House Shape (8 points)

A house-shaped figure consists of a 5 cm by 4 cm rectangle with a triangle on top having a base of 5 cm and height of 3 cm. Find the total area. (Note: The triangle is not rectangular, so use the triangle formula: Area = (base × height) ÷ 2)

Solution:

Rectangle: 5 × 4 = 20 cm²

Triangle: (5 × 3) ÷ 2 = 7.5 cm²

Total Area = 20 + 7.5 = 27.5 cm²

6 Real World Application (6 points)

A room is shaped like an L with one section measuring 6 m by 4 m and another section measuring 3 m by 4 m. How much carpet is needed to cover the floor?

Solution:

First rectangle: 6 × 4 = 24 m²

Second rectangle: 3 × 4 = 12 m²

Total Area = 24 + 12 = 36 m²

Carpet needed: 36 m²

7 Challenge Problem (8 points)

A rectangular garden measures 12 feet by 8 feet. A path 2 feet wide runs along the inside edge of the garden all around. What is the area of the path?

Solution:

Total garden area: 12 × 8 = 96 ft²

Inner rectangle area: (12-4) × (8-4) = 8 × 4 = 32 ft²

Path area = 96 - 32 = 64 ft²

Questions & Answers

Question: How do I know where to break apart a composite rectangular shape into simpler rectangles?

Answer: Great question! Here are strategies to identify where to separate:

  • Look for straight edges: Most composite rectangular shapes are made by combining rectangles
  • Find right angles: These often indicate where one rectangle ends and another begins
  • Look for common dimensions: If two parts share the same length or width, that's likely a separation line
  • Try different ways: Sometimes there are multiple correct ways to break apart a figure

Remember: The goal is to create rectangles whose areas you can easily calculate using the formula A = Length × Width!

Question: What if a shape has a part cut out of it? Do I still add the areas?

Answer: No, when a part is cut out or removed, you subtract its area instead of adding it!

  • When shapes are joined: Add the areas together
  • When parts are removed: Subtract the removed area from the whole
  • Formula: Remaining Area = Original Area - Removed Area

Think of it like cutting a piece out of paper. You start with the full sheet (large area) and remove a piece (subtract the small area).

Example: If you have a 10×8 rectangle (area = 80) and cut out a 2×3 rectangle (area = 6), the remaining area is 80 - 6 = 74 square units.

Question: Can composite shapes have other shapes besides rectangles?

Answer: Yes! Composite shapes can include triangles, circles, and other polygons. However, in grade 4, we focus on composite rectangular shapes because rectangles are easier to work with using the formula A = Length × Width.

  • Triangles: Area = (base × height) ÷ 2
  • Squares: Area = Side × Side
  • Other polygons: May require more advanced formulas

For now, focus on rectangles since they're the foundation for understanding area. As you advance, you'll learn about other shapes too!