Solved Exercises on Area Word Problems in Grade 3

Master area word problems: composite shapes, rectangles, and complex area problems through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Composite Shape Area
Exercise 1
Sarah has a garden made of two rectangles. The first rectangle is 8 feet long and 4 feet wide. The second rectangle is attached to the first and is 5 feet long and 4 feet wide. What is the total area of the garden?
Definition:

Composite shape: A shape made up of two or more basic shapes (rectangles, squares, triangles).

Area: The amount of space inside a shape, measured in square units.

Area calculation method:
  1. Identify the individual shapes that make up the composite shape
  2. Calculate the area of each individual shape
  3. Add all the areas together to find the total area
Shape 1
8 × 4 = 32 ft²
Shape 2
5 × 4 = 20 ft²
Total Area
32 + 20 = 52 ft²
Step 1: Identify the shapes

The garden consists of two rectangles: Rectangle 1 and Rectangle 2

Step 2: Calculate area of Rectangle 1

Length = 8 feet, Width = 4 feet

Area = Length × Width = 8 × 4 = 32 square feet

Step 3: Calculate area of Rectangle 2

Length = 5 feet, Width = 4 feet

Area = Length × Width = 5 × 4 = 20 square feet

Step 4: Add the areas together

Total Area = Area of Rectangle 1 + Area of Rectangle 2

Total Area = 32 + 20 = 52 square feet

The total area of the garden is 52 square feet.
Final answer:

The total area of the garden is 52 square feet.

Applied rules:

• Area of rectangle: Length × Width

• Composite area: Sum of individual areas

• Units: Area is measured in square units (ft²)

2 Room Area Problem
Exercise 2
A house has a living room that is 12 feet long and 10 feet wide. There is also a kitchen that is 8 feet long and 6 feet wide. If the dining room is 9 feet long and 7 feet wide, what is the total area of these three rooms?
Definition:

Multiple rectangles: Finding the combined area of several rectangular spaces.

Living Room
12 × 10 = 120 ft²
Kitchen
8 × 6 = 48 ft²
Dining Room
9 × 7 = 63 ft²
Total Area
120 + 48 + 63 = 231 ft²
Step 1: Calculate Living Room Area

Length = 12 feet, Width = 10 feet

Area = 12 × 10 = 120 square feet

Step 2: Calculate Kitchen Area

Length = 8 feet, Width = 6 feet

Area = 8 × 6 = 48 square feet

Step 3: Calculate Dining Room Area

Length = 9 feet, Width = 7 feet

Area = 9 × 7 = 63 square feet

Step 4: Add all areas together

Total Area = 120 + 48 + 63 = 231 square feet

The total area of all three rooms is 231 square feet.
Final answer:

The total area of all three rooms is 231 square feet.

Applied rules:

• Area of rectangle: Length × Width

• Multiple areas: Add individual areas together

• Consistency: Use the same units for all measurements

3 Area Comparison Problem
Exercise 3
Mike has a rectangular garden that is 15 feet long and 8 feet wide. His neighbor has a square garden with sides of 11 feet. Whose garden has a larger area, and by how much?
Definition:

Area comparison: Calculating areas of different shapes and comparing them.

Area of square: Side × Side = Side²

Mike's Garden
15 × 8 = 120 ft²
Neighbor's Garden
11 × 11 = 121 ft²
Difference
121 - 120 = 1 ft²
Step 1: Calculate Mike's garden area

Rectangle: Length = 15 feet, Width = 8 feet

Area = 15 × 8 = 120 square feet

Step 2: Calculate neighbor's garden area

Square: Side = 11 feet

Area = 11 × 11 = 121 square feet

Step 3: Compare the areas

Neighbor's garden: 121 ft²

Mike's garden: 120 ft²

Difference: 121 - 120 = 1 square foot

Step 4: Determine the answer

The neighbor's garden is larger by 1 square foot.

The neighbor's garden is larger by 1 square foot.
Final answer:

The neighbor's garden is larger by 1 square foot.

Applied rules:

• Area of rectangle: Length × Width

• Area of square: Side × Side

• Comparison: Subtract smaller area from larger area

Area Formulas and Methods
\(Area = Length × Width\)
Area of Rectangle
Rectangle
\(A = l × w\)
Length × Width
Square
\(A = s × s\)
Side × Side
Composite Shape
\(A = A_1 + A_2 + ...\)
Sum of Individual Areas
Key definitions:

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

Perimeter: The distance around the edge of a shape.

Composite shape: A shape made up of two or more basic shapes.

Complete methodology:
  1. Read the problem carefully: Identify what shapes are involved
  2. Find dimensions: Note the length and width of each shape
  3. Apply area formula: Use the appropriate area formula
  4. Calculate: Perform the multiplication
  5. Check units: Ensure answer is in square units
Tip 1: Draw a picture to visualize the problem.
Tip 2: Always include units in your final answer.
Tip 3: For composite shapes, break them into simpler shapes first.
Tip 4: Double-check your multiplication facts.
Common errors: Forgetting to multiply length by width, mixing up length and width, forgetting square units.
Key notes: Area is always measured in square units (cm², m², ft², etc.), area measures the inside of a shape.
Formulas to know by heart:

• Rectangle: Area = Length × Width

• Square: Area = Side × Side

• Composite Shape: Area = Sum of individual areas

• Area is always expressed in square units

Solution: Exercises 4 to 5
4 Missing Dimension Problem
Exercise 4
A rectangular playground has an area of 144 square feet. If the width is 9 feet, what is the length of the playground?
Definition:

Missing dimension: Using the area formula to find an unknown length or width.

Area formula rearrangement: If Area = Length × Width, then Length = Area ÷ Width

Given
Area = 144 ft², Width = 9 ft
Formula
Length = Area ÷ Width
Calculation
144 ÷ 9 = 16 ft
Step 1: Identify known values

Area = 144 square feet, Width = 9 feet, Length = ?

Step 2: Use area formula

Area = Length × Width

144 = Length × 9

Step 3: Solve for Length

To find Length, divide both sides by Width

Length = Area ÷ Width

Length = 144 ÷ 9 = 16 feet

Step 4: Verify the answer

Check: Length × Width = 16 × 9 = 144 ft² ✓

The length of the playground is 16 feet.
Final answer:

The length of the playground is 16 feet.

Applied rules:

• Area formula: Area = Length × Width

• Rearranging formula: Length = Area ÷ Width

• Division: Divide area by known dimension

5 L-Shaped Area Problem
Exercise 5
A room has an L-shaped floor plan. One part of the room is 10 feet long and 6 feet wide. The other part is 4 feet long and 6 feet wide, attached to the first part along the 6-foot side. What is the total area of the room?
Definition:

L-shaped figure: A composite shape that looks like the letter 'L', made up of two rectangles.

Decomposition: Breaking a complex shape into simpler shapes.

Part 1
10 × 6 = 60 ft²
Part 2
4 × 6 = 24 ft²
Total Area
60 + 24 = 84 ft²
Step 1: Break the L-shape into rectangles

Rectangle 1: 10 feet long × 6 feet wide

Rectangle 2: 4 feet long × 6 feet wide

Step 2: Calculate area of Rectangle 1

Area = Length × Width = 10 × 6 = 60 square feet

Step 3: Calculate area of Rectangle 2

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

Step 4: Add the areas together

Total Area = Area of Rectangle 1 + Area of Rectangle 2

Total Area = 60 + 24 = 84 square feet

Step 5: Verify the answer

Check: The L-shape is made up of two rectangles with areas 60 ft² and 24 ft²

Total: 60 + 24 = 84 ft² ✓

The total area of the room is 84 square feet.
Final answer:

The total area of the room is 84 square feet.

Applied rules:

• Decomposition: Break complex shapes into simpler ones

• Area of rectangle: Length × Width

• Composite area: Sum of individual areas

Area Laws, Methods, Rules, and Definitions
\(Area = Length × Width\)
Area Formula
Key definitions:

Area: The measure of the space inside a two-dimensional shape, expressed in square units.

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

Square: A rectangle with all sides equal.

Composite shape: A shape formed by combining two or more basic shapes.

Complete methodology:
  1. Identify the shape(s): Determine if it's a rectangle, square, or composite shape
  2. Measure dimensions: Find the length and width of each shape
  3. Select formula: Use the appropriate area formula
  4. Calculate: Multiply length by width
  5. Check units: Express answer in square units
  6. Verify: Double-check calculations
Tip 1: Draw a diagram to visualize the problem.
Tip 2: Label all dimensions on your diagram.
Tip 3: For composite shapes, find the area of each part separately.
Tip 4: Always write the unit of measurement (ft², cm², etc.).
Tip 5: Check if your answer makes sense in the context of the problem.
Common errors: Confusing area with perimeter, forgetting to multiply, using wrong units, miscalculating.
Exam preparation: Practice finding area of rectangles, squares, and composite shapes; memorize formulas; work on word problems.
Real-world applications: Carpeting rooms, painting walls, gardening, construction projects.
Formulas to know by heart:

• Rectangle: Area = Length × Width

• Square: Area = Side × Side

• Composite Shape: Total Area = Sum of Individual Areas

• To find missing dimension: Length = Area ÷ Width (or vice versa)

Visualizing Area Concepts
Understanding Area Relationships
Explore how changing dimensions affects area:
Rectangle A: 6 × 4 = 24 sq units
Rectangle B: 8 × 3 = 24 sq units
Rectangle C: 12 × 2 = 24 sq units

Key Insight: Different dimensions can result in the same area!

  • All three rectangles have the same area (24 sq units)
  • This demonstrates that multiple length-width combinations can produce the same area
  • Understanding this helps solve problems where one dimension is unknown

Problem-Solving Strategy: When given the area and one dimension, use division to find the missing dimension.

Questions & Answers

Question: I don't understand why we multiply length by width to find area. Why isn't it length plus width?

Answer: Great question! Think of area as counting the number of square units that fit inside a shape:

  • If you have a rectangle that is 3 units long and 2 units wide, imagine covering it with tiny 1×1 squares
  • You can fit 3 squares along the length
  • You can stack 2 rows of these squares along the width
  • Total squares = 3 squares in each row × 2 rows = 6 squares

So area = length × width because we're counting how many squares fit in both directions. Adding length + width would just give us the total of those two measurements, not the space inside.

Try drawing a 3×2 rectangle on graph paper and count the squares - you'll see there are 6 squares inside!

Question: My child is struggling with composite shapes. How can I help them break down complex area problems?

Answer: Here's a helpful strategy for composite shapes:

  • Step 1: Have your child draw lines to separate the complex shape into simpler rectangles
  • Step 2: Label the dimensions of each rectangle they've identified
  • Step 3: Calculate the area of each rectangle separately
  • Step 4: Add all the individual areas together

Use hands-on materials like colored paper rectangles that can be physically arranged to form composite shapes. This helps children visualize how the parts combine to form the whole.

Practice with real-world examples like floor plans of rooms or garden layouts to make it more meaningful.

Question: How do I know if a problem is asking for area or perimeter? They seem similar.

Answer: Here's how to tell the difference:

  • Area problems ask about: "How much space?", "Covering", "Inside", "Floor", "Garden", "Carpet", "Painting a wall"
  • Perimeter problems ask about: "Fencing", "Border", "Around", "Edge", "Distance around", "Walking around"
  • Look for keywords: Area uses words like "square", "covering", "space", while perimeter uses "around", "fence", "border"

Think of it this way: Area is like spreading a blanket over the shape (inside), and perimeter is like walking around the edge of the shape (outside).

Area answers end with square units (ft², cm²), while perimeter answers end with regular units (ft, cm).