Solved Exercises on Area of Rectangles in Grade 4

Master rectangle area: calculating area using length and width through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic rectangle area
Exercise 1
A rectangle has a length of 8 meters and a width of 5 meters.
What is the area of the rectangle?
Definition:

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Rectangle area method:
  1. Identify the length and width of the rectangle
  2. Use the formula: Area = length × width
  3. Substitute the known values
  4. Calculate the result
  5. Include the square units (m²)
📏
Length
8m
×
📐
Width
5m
=
40m²
Area
Step 1: Identify the dimensions

Length = 8 meters, Width = 5 meters

Step 2: Apply the rectangle area formula

Area = length × width

Step 3: Substitute the values

Area = 8 × 5

Step 4: Calculate the result

Area = 40 square meters

Step 5: Include the correct units

Area = 40 m²

Area = 40 m²
Final answer:

The area of the rectangle is 40 square meters.

Applied rules:

• Rectangle area: A = l × w

• Square units: Area is always in square units (m², cm², ft²)

2 Rectangle with different units
Exercise 2
A rectangular picture frame has a length of 12 inches and a width of 8 inches.
What is the area of the picture inside the frame?
Definition:

Square units: Units for area that represent two-dimensional space (in², cm², m²)

🖼️
Picture
12in × 8in
→
A=12×8
Area calculation
→
96in²
Area
Step 1: Identify the dimensions

Length = 12 inches, Width = 8 inches

Step 2: Apply the rectangle area formula

Area = length × width

Step 3: Substitute the values

Area = 12 × 8

Step 4: Calculate the result

Area = 96 square inches

Step 5: Include the correct units

Area = 96 in²

Area = 96 in²
Final answer:

The area of the picture inside the frame is 96 square inches.

Applied rules:

• Rectangle area: A = l × w

• Square units: Area is always in square units

3 Square as a rectangle
Exercise 3
A square garden has sides that are each 6 feet long.
What is the area of the garden? (Note: A square is a special rectangle)
Definition:

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

🟦
Square
All sides = 6ft
→
A=6×6
Rectangle formula
→
36ft²
Area
Step 1: Identify the dimensions

Length = 6 feet, Width = 6 feet (since it's a square)

Step 2: Apply the rectangle area formula

Area = length × width

Step 3: Substitute the values

Area = 6 × 6

Step 4: Calculate the result

Area = 36 square feet

Step 5: Alternative calculation

Since all sides are equal: Area = 6² = 36 ft²

Area = 36 ft²
Final answer:

The area of the square garden is 36 square feet.

Applied rules:

• Square as rectangle: A = l × w where l = w

• Special case: A = s² for squares

Rectangle Area Formulas and Rules
A = l × w
Rectangle Area Formula
Standard Formula
A = l × w
Length × Width
Square Formula
A = s²
Side × Side
Area Units
Square Units
m², cm², ft², in²
Key definitions:

Area: The amount of space inside a two-dimensional shape

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Length: The longer side of a rectangle

Width: The shorter side of a rectangle

Square: A rectangle with all four sides equal in length

Square units: Units that measure two-dimensional space (m², cm², ft², in²)

Rectangle area calculation methods:
  1. Standard method: A = length × width
  2. Special case: For squares, A = side × side = side²
  3. Verification: Double-check by counting squares or estimating
  4. Unit consistency: Ensure all measurements use the same units
  5. Include units: Always add square units to your answer
Key Property: Rectangle has opposite sides equal in length.
Key Property: Area is always measured in square units.
Detailed Summary:

Area of Rectangles - Complete Guide

Definitions:

  • Rectangle: Four-sided shape with opposite sides equal and all angles 90°
  • Area: Amount of space inside a shape
  • Length: Longer side of rectangle
  • Width: Shorter side of rectangle
  • Square units: Units for measuring area (m², cm², ft²)

Core Rules:

  • Rectangle area: A = length × width
  • Square area: A = side × side = side²
  • Always use consistent units of measurement
  • Area is always in square units

Methods:

  • Identify length and width
  • Apply appropriate formula
  • Substitute known values
  • Calculate the result
  • Include square units in the answer

Examples:

  • Rectangle: 8m × 5m → A = 8×5 = 40m²
  • Rectangle: 12in × 8in → A = 12×8 = 96in²
  • Square: 6ft sides → A = 6×6 = 36ft²

Tips & Tricks:

  • Always include square units (m², ft², in²) in your answer
  • Remember: area measures the inside space
  • Make sure all measurements use the same units
  • Draw a diagram to visualize the rectangle

Common Mistakes:

  • Forgetting to include square units
  • Mixing different units of measurement
  • Using perimeter formula instead of area
  • Confusing area (space inside) with perimeter (distance around)

Memorization Aids:

  • "Area is inside, Perimeter is around"
  • "Length times width for rectangles"
  • "Side times side for squares"
  • "Square units for area"
Solution: Exercises 4 to 5
4 Rectangle area with relationship
Exercise 4
A rectangular room has a width of 7 feet. The length is 3 feet more than the width.
What is the area of the room?
Definition:

Relationship problem: A problem where dimensions are related by a mathematical relationship

7ft
Width
→
10ft
Length
Width + 3ft
→
A=10×7
Formula
→
70ft²
Area
Step 1: Identify the known dimension

Width = 7 feet

Step 2: Find the length using the relationship

Length = Width + 3 = 7 + 3 = 10 feet

Step 3: Apply the rectangle area formula

Area = length × width

Step 4: Substitute the values

Area = 10 × 7

Step 5: Calculate the result

Area = 70 square feet

Area = 70 ft²
Final answer:

The area of the room is 70 square feet.

Applied rules:

• Relationship solving: Use given relationship to find unknown dimension

• Rectangle formula: A = l × w

5 Carpet installation problem
Exercise 5
A rectangular living room measures 15 feet long and 10 feet wide.
How many square feet of carpet are needed to cover the entire floor?
Definition:

Application problem: A real-world problem that uses mathematical concepts

🪵
Room
15ft × 10ft
→
A=15×10
Carpet needed
→
150ft²
Carpet
Step 1: Identify the dimensions

Length = 15 feet, Width = 10 feet

Step 2: Apply the rectangle area formula

Area = length × width

Step 3: Substitute the values

Area = 15 × 10

Step 4: Calculate the result

Area = 150 square feet

Step 5: Interpret the answer

150 square feet of carpet are needed

Carpet needed: 150 ft²
Final answer:

150 square feet of carpet are needed to cover the floor.

Applied rules:

• Real-world application: Area equals amount of material needed

• Rectangle formula: A = l × w

Advanced Rectangle Area Applications and Problem Solving
A = l × w
Rectangle Area Formula
Key definitions:

Area: The amount of space inside a two-dimensional shape

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Square units: Units that measure two-dimensional space (m², cm², ft², in²)

Application problem: A real-world problem that requires mathematical calculation

Relationship problem: A problem where dimensions are connected by a mathematical relationship

Complete problem-solving methodology:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Identify: Determine the rectangle's length and width
  3. Check: Verify that all measurements use the same units
  4. Apply: Use the area formula A = l × w
  5. Calculate: Perform the mathematical operations
  6. Verify: Check that the answer makes sense in context
Tip 1: Always use the same units for all measurements in a problem.
Tip 2: Draw a simple diagram to visualize the rectangle and label the sides.
Tip 3: Remember: area measures the inside space, perimeter measures the outside.
Tip 4: Always include square units (m², ft², in²) in your final answer.
Common errors: Forgetting square units, mixing units, using perimeter formula instead of area.
Exam preparation: Practice with various units and relationship problems.
Properties to know by heart:

• Rectangle: A = l × w

• Square: A = s²

• Area is always in square units

• Opposite sides of rectangle are equal

• Always check for consistent units

Extended Learning Summary:

Advanced Rectangle Area Concepts

Problem-Solving Strategies:

  • Diagram drawing: Visualize the rectangle with a simple sketch
  • Unit consistency: Ensure all measurements use the same units
  • Formula selection: Use A = l × w for rectangles
  • Verification: Double-check calculations using alternative methods

Real-world Applications:

  • Flooring: Calculating carpet or tile needed for rectangular rooms
  • Gardening: Determining space for rectangular garden beds
  • Painting: Calculating paint needed for rectangular walls
  • Landscaping: Measuring area for rectangular patios or decks

Advanced Problem Types:

  • Relationship problems with proportional dimensions
  • Missing dimension problems where area is known
  • Unit conversion problems requiring measurement changes
  • Composite problems combining area with other concepts

Questions & Answers

Question: I sometimes mix up length and width. Does it matter which is which in the formula?

Answer: No, it doesn't matter which side you call length and which you call width! The formula A = l × w will give the same result regardless:

  • As long as you identify both different measurements
  • Then multiply them together
  • For example: if sides are 5 and 8, A = 5 × 8 = 40, same as A = 8 × 5 = 40

Just make sure you multiply both dimensions together!

Question: Can I use the rectangle area formula for a square?

Answer: Yes, absolutely! A square is a special type of rectangle:

  • A square has all the properties of a rectangle (opposite sides equal, all angles 90°)
  • Plus, all sides are equal in a square
  • So A = l × w becomes A = s × s = s² for squares
  • Both formulas work, but A = s² is simpler for squares

All squares are rectangles, but not all rectangles are squares!

Question: What if I only know the area and one dimension? Can I find the other dimension?

Answer: Yes! If you know the area and one dimension, you can find the other:

  • Start with the formula: A = l × w
  • Rearrange to find the unknown: l = A ÷ w or w = A ÷ l
  • Divide the area by the known dimension
  • For example: if A = 24 and l = 6, then w = 24 ÷ 6 = 4

This is called a "missing dimension" problem!

Question: Why do we use square units for area?

Answer: We use square units because area measures two-dimensional space:

  • Length: One-dimensional (measured in meters, feet)
  • Area: Two-dimensional (length × width, so meters × meters = square meters)
  • Volume: Three-dimensional (length × width × height, cubic units)

Area involves multiplying two dimensions, so we get square units!

Question: How is the area of a rectangle different from its perimeter?

Answer: Area and perimeter measure different aspects of a rectangle:

  • Area: Space inside the rectangle (square units like m², ft²)
  • Perimeter: Distance around the outside (linear units like m, ft)
  • Think of area as the grass inside a yard, perimeter as the fence around it
  • Rectangle area: A = l × w, Rectangle perimeter: P = 2(l + w)

They're both measurements of rectangles but for different purposes!