Solved Exercises on Area Problems in Grade 3

Master area calculations: rectangles, squares, and composite figures through these 5 detailed exercises with visual explanations.

Solution: Exercises 1 to 3
1 Rectangle Area
Exercise 1
Find the area of a rectangle that is 5 units long and 3 units wide.
Definition:

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

Rectangle: A four-sided shape with four right angles.

Area calculation method:
  1. Identify the length and width of the rectangle
  2. Apply the formula: Area = Length × Width
  3. Count the number of square units inside the shape
  4. Write the answer with square units
Length
5 units
Width
3 units
Area
15 sq units
Step 1: Identify dimensions

Length = 5 units, Width = 3 units

Step 2: Apply the formula

Area = Length × Width = 5 × 3

Step 3: Calculate the area

5 × 3 = 15 square units

Step 4: Count the squares

We can count 15 squares inside the rectangle

Area = 15 square units
Final answer:

The area of the rectangle is 15 square units.

Applied rules:

• Formula: Area of rectangle = Length × Width

• Units: Area is always measured in square units

• Visualization: Counting squares confirms our calculation

2 Square Area
Exercise 2
Find the area of a square with sides of 4 units.
Definition:

Square: A rectangle where all four sides are equal in length.

Formula: Area of square = Side × Side = Side²

Side length
4 units
Formula
Side²
Area
16 sq units
Step 1: Identify the side length

All sides of the square = 4 units

Step 2: Apply the formula

Area = Side × Side = 4 × 4

Step 3: Calculate the area

4 × 4 = 16 square units

Step 4: Verify by counting

We can count 16 squares inside the square

Area = 16 square units
Final answer:

The area of the square is 16 square units.

Applied rules:

• Formula: Area of square = Side × Side = Side²

• Equality: All sides of a square are equal

• Squaring: Multiplying a number by itself

3 Comparing Areas
Exercise 3
Which has a larger area: a rectangle that is 6 units long and 2 units wide, or a square with sides of 3 units?
Definition:

Comparison: Finding which of two areas is greater by calculating each one.

Rectangle
6×2=12
Square
3²=9
Comparison
12 > 9
Step 1: Calculate rectangle area

Area = Length × Width = 6 × 2 = 12 square units

Step 2: Calculate square area

Area = Side × Side = 3 × 3 = 9 square units

Step 3: Compare the areas

12 > 9, so the rectangle has a larger area

Step 4: State the conclusion

The rectangle with dimensions 6×2 has a larger area than the 3×3 square

Rectangle (12 sq units) > Square (9 sq units)
Final answer:

The rectangle with dimensions 6 units by 2 units has a larger area (12 square units) than the square with sides of 3 units (9 square units).

Applied rules:

• Calculate both areas: Before comparing, find each area separately

• Comparison: Use >, <, or = to compare numerical values

• Units: Both areas must be in the same units for comparison

Area Rules and Methods
Area = Length × Width
Rectangle Area
Square Area
Area = Side²
All sides equal
Rectangle Area
Area = Length × Width
Opposite sides equal
Unit of Measurement
Square Units
Always write as sq units
Key definitions:

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

Square unit: A square with sides of 1 unit length

Dimensions: The measurements of a shape (length, width, height)

Area calculation methodology:
  1. Identify the shape: Rectangle, square, or composite figure
  2. Measure dimensions: Find length, width, or side measurements
  3. Apply the formula: Use the appropriate area formula
  4. Calculate: Perform the multiplication
  5. Label: Always include square units in your answer
Tip 1: Count the squares inside a shape to verify your area calculation.
Tip 2: A square is a special type of rectangle where length equals width.
Tip 3: Always write area answers with square units (sq units, cm², m², etc.).
Tip 4: For composite figures, break them into rectangles and add the areas.
Common mistakes: Forgetting to include square units, confusing area with perimeter, miscounting dimensions.
Memorization trick: "Area is Length Times Width" - ALTW sounds like "Always Try Working".
Area formulas to know:

• Rectangle: Area = Length × Width

• Square: Area = Side × Side = Side²

• Composite figure: Add areas of individual rectangles/squares

• Unit conversion: 1 ft² = 144 in², 1 m² = 10,000 cm²

Solution: Exercises 4 to 5
4 Composite Figure
Exercise 4
Find the area of this L-shaped figure made of two rectangles: one is 4 units by 2 units, and the other is 3 units by 2 units.
Definition:

Composite figure: A shape made up of two or more simpler shapes like rectangles or squares.

Part 1
4×2=8
Part 2
3×2=6
Total
8+6=14
Step 1: Break into rectangles

Rectangle 1: 4 units × 2 units

Rectangle 2: 3 units × 2 units

Step 2: Calculate area of first rectangle

Area₁ = 4 × 2 = 8 square units

Step 3: Calculate area of second rectangle

Area₂ = 3 × 2 = 6 square units

Step 4: Add the areas together

Total Area = Area₁ + Area₂ = 8 + 6 = 14 square units

Total Area = 14 square units
Final answer:

The area of the L-shaped figure is 14 square units.

Applied rules:

• Decomposition: Break complex shapes into simpler ones

• Addition: Sum the areas of individual parts

• Verification: Count total squares to confirm answer

5 Real-world Application
Exercise 5
Sarah wants to carpet her rectangular bedroom. The room is 8 feet long and 6 feet wide. How many square feet of carpet does she need?
Definition:

Real-world application: Using area calculations to solve practical problems involving coverage or measurement.

Length
8 ft
Width
6 ft
Area
48 sq ft
Step 1: Identify the dimensions

Length = 8 feet, Width = 6 feet

Step 2: Apply the area formula

Area = Length × Width = 8 × 6

Step 3: Calculate the area

8 × 6 = 48 square feet

Step 4: Interpret the result

Sarah needs 48 square feet of carpet to cover the entire floor

Step 5: Check units

Since measurements were in feet, the area is in square feet

Sarah needs 48 square feet of carpet
Final answer:

Sarah needs 48 square feet of carpet to cover her bedroom floor.

Applied rules:

• Real-world context: Area tells us how much material is needed to cover a surface

• Unit consistency: Answer units match the given measurement units

• Practical application: Area calculations are essential for construction and design

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.

Square unit: A unit of area equal to a square measuring one unit on each side.

Dimension: The measurement of length, width, or height of a geometric figure.

Perimeter vs Area: Perimeter measures distance around a shape; area measures space inside.

Complete area calculation methodology:
  1. Identify the shape: Determine if it's a rectangle, square, triangle, or composite figure
  2. Measure dimensions: Use rulers or given measurements to find necessary lengths
  3. Select the formula: Choose the appropriate area formula for the shape
  4. Substitute values: Plug the measured dimensions into the formula
  5. Calculate: Perform the mathematical operations
  6. Verify: Count squares or use alternative method to check accuracy
  7. Label: Express the answer in square units
Tip 1: Draw a grid inside the shape to visualize square units and verify your calculation.
Tip 2: For rectangles, it doesn't matter which dimension you call length or width.
Tip 3: Squares are rectangles with equal sides, so you can use either formula.
Tip 4: When adding areas of composite figures, make sure not to double-count overlapping regions.
Tip 5: In word problems, identify keywords like "cover," "carpet," "paint," or "floor" to know you need area.
Common errors: Confusing area with perimeter, forgetting square units, misreading dimensions.
Memory aids: "Area is Inside" - AI, "Length Times Width" - LTW, "All Things Work" - ATW.
Visual strategy: Draw the shape and label dimensions before calculating.
Essential area formulas and rules:

• Rectangle Area: Area = Length × Width

• Square Area: Area = Side × Side = Side²

• Unit Consistency: All dimensions must be in the same unit before multiplying

• Square Units: Area answers always include squared units (ft², m², cm², etc.)

• Composite Figures: Add areas of non-overlapping parts

• Verification: Counting unit squares confirms calculated area

Questions & Answers

Question: My students are struggling to understand why area is measured in square units. How can I help them grasp this concept better?

Answer: Great question! The concept of square units is fundamental to understanding area. Here are effective teaching strategies:

  • Concrete materials: Use square tiles or graph paper to physically fill shapes. Students can count the squares to see area directly.
  • Visual demonstration: Show how a "square unit" is literally a square with sides of 1 unit length. Area measures how many of these squares fit inside a shape.
  • Hands-on activities: Have students measure different rectangles using square tiles and then multiply length by width to see the connection.
  • Real-world examples: Relate to covering surfaces - carpeting rooms, tiling floors, painting walls.

Example activity: Give students a 3×4 rectangle and ask them to fill it with 1-inch square tiles. They'll discover it takes 12 tiles, reinforcing that area = 3×4 = 12 square inches.

Question: I sometimes get confused between area and perimeter. How do I remember which is which?

Answer: This is a very common confusion! Here are memory tricks to keep them straight:

  • Area = Inside: AREA starts with "A" and INSIDE starts with "I" - Area measures the space INSIDE a shape.
  • Perimeter = Border: PERIMETER starts with "P" and BORDER starts with "B" - Perimeter measures the BORDER around a shape.
  • Memory phrase: "Perimeter is the fence around the yard; Area is the grass inside."
  • Visual clue: Draw a rectangle and shade the inside for area; draw a line around the edge for perimeter.

Example: For a 4×3 rectangle, area = 4×3 = 12 sq units (the inside), while perimeter = 4+3+4+3 = 14 units (the border).

Question: My child is having trouble with composite figures. What's the best way to approach these area problems?

Answer: Composite figures can seem challenging, but breaking them down makes them manageable:

  • Step 1 - Identify parts: Look for familiar shapes within the composite figure (rectangles, squares).
  • Step 2 - Separate the figure: Mentally or physically draw lines to separate the composite into distinct rectangles.
  • Step 3 - Calculate each area: Find the area of each individual rectangle using the standard formula.
  • Step 4 - Add the areas: Sum all individual areas to get the total area.

Practice tip: Start with simple L-shapes, then progress to more complex figures. Encourage drawing the separation lines to visualize the individual rectangles.

Example: An L-shape made of a 4×2 rectangle and a 2×3 rectangle has total area = (4×2) + (2×3) = 8 + 6 = 14 square units.