Solved Exercises on Perimeter of Rectangles in Grade 4

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

Solution: Exercises 1 to 3
1 Basic rectangle perimeter
Exercise 1
A rectangle has a length of 10 meters and a width of 6 meters.
What is the perimeter of the rectangle?
Definition:

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

Rectangle perimeter method:
  1. Identify the length and width of the rectangle
  2. Use the formula: Perimeter = 2 × (length + width)
  3. Substitute the known values
  4. Calculate inside the parentheses first
  5. Complete the multiplication
📏
Length
10m
+
📐
Width
6m
×
2
Formula
=
32m
Perimeter
Step 1: Identify the dimensions

Length = 10 meters, Width = 6 meters

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Step 3: Substitute the values

Perimeter = 2 × (10 + 6)

Step 4: Calculate inside the parentheses

Perimeter = 2 × 16

Step 5: Complete the multiplication

Perimeter = 32 meters

Perimeter = 32 meters
Final answer:

The perimeter of the rectangle is 32 meters.

Applied rules:

• Rectangle perimeter: P = 2 × (l + w)

• Order of operations: Parentheses first, then multiplication

2 Rectangle with different units
Exercise 2
A rectangular picture frame has a length of 24 inches and a width of 18 inches.
What is the perimeter of the frame?
Definition:

Consistent units: All measurements must use the same unit of measurement

🖼️
Frame
24in × 18in
→
P=2(24+18)
Formula
→
84in
Perimeter
Step 1: Identify the dimensions

Length = 24 inches, Width = 18 inches

Step 2: Verify unit consistency

Both measurements are in inches, so units are consistent

Step 3: Apply the perimeter formula

Perimeter = 2 × (length + width)

Step 4: Substitute the values

Perimeter = 2 × (24 + 18)

Step 5: Calculate the result

Perimeter = 2 × 42 = 84 inches

Perimeter = 84 inches
Final answer:

The perimeter of the picture frame is 84 inches.

Applied rules:

• Unit consistency: All measurements must use the same unit

• Rectangle formula: P = 2 × (l + w)

3 Square as a special rectangle
Exercise 3
A square garden has sides that are each 12 feet long.
What is the perimeter 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 = 12ft
→
P=2(12+12)
Rectangle formula
→
48ft
Perimeter
Step 1: Identify the dimensions

Length = 12 feet, Width = 12 feet (since it's a square)

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Step 3: Substitute the values

Perimeter = 2 × (12 + 12)

Step 4: Calculate the result

Perimeter = 2 × 24 = 48 feet

Step 5: Alternative calculation

Since all sides are equal: Perimeter = 4 × 12 = 48 feet

Perimeter = 48 feet
Final answer:

The perimeter of the square garden is 48 feet.

Applied rules:

• Square as rectangle: P = 2 × (l + w) where l = w

• Special case: P = 4 × side length for squares

Rectangle Perimeter Formulas and Rules
P = 2(l + w)
Rectangle Perimeter Formula
Standard Formula
P = 2(l + w)
2 × (length + width)
Alternative Formula
P = l + w + l + w
Add all sides
Square Formula
P = 4s
4 × side length
Key definitions:

Perimeter: The total distance around the outside of a 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

Formula: A mathematical rule expressed using symbols

Rectangle perimeter calculation methods:
  1. Standard method: P = 2 × (length + width)
  2. Addition method: P = length + width + length + width
  3. Special case: For squares, P = 4 × side length
  4. Verification: Double-check by adding all sides separately
  5. Unit consistency: Ensure all measurements use the same units
Key Property: Rectangle has opposite sides equal in length.
Key Property: Perimeter is always measured in linear units.
Detailed Summary:

Perimeter of Rectangles - Complete Guide

Definitions:

  • Rectangle: Four-sided shape with opposite sides equal and all angles 90°
  • Perimeter: Distance around the outside of a shape
  • Length: Longer side of rectangle
  • Width: Shorter side of rectangle
  • Formula: Mathematical rule for calculations

Core Rules:

  • Rectangle perimeter: P = 2 × (length + width)
  • Alternative: P = l + w + l + w
  • Square perimeter: P = 4 × side length
  • Always use consistent units of measurement

Methods:

  • Identify length and width
  • Apply appropriate formula
  • Substitute known values
  • Calculate following order of operations

Examples:

  • Rectangle: 10m × 6m → P = 2(10+6) = 32m
  • Rectangle: 24in × 18in → P = 2(24+18) = 84in
  • Square: 12ft sides → P = 4×12 = 48ft

Tips & Tricks:

  • Always check that you're using the same units
  • Remember: length and width are used only once each in the formula
  • For squares, you can use the special formula P = 4s
  • Draw a diagram to visualize the rectangle

Common Mistakes:

  • Mixing different units of measurement
  • Forgetting to multiply by 2 in the formula
  • Counting length and width twice instead of once each
  • Using area formula instead of perimeter

Memorization Aids:

  • "2 times length plus width"
  • "Add length and width, then double it"
  • Square: "4 times any side"
Solution: Exercises 4 to 5
4 Rectangle perimeter with relationship
Exercise 4
A rectangular room has a width of 8 feet. The length is 3 feet more than the width.
What is the perimeter of the room?
Definition:

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

8ft
Width
→
11ft
Length
Width + 3ft
→
P=2(11+8)
Formula
→
38ft
Perimeter
Step 1: Identify the known dimension

Width = 8 feet

Step 2: Find the length using the relationship

Length = Width + 3 = 8 + 3 = 11 feet

Step 3: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Step 4: Substitute the values

Perimeter = 2 × (11 + 8)

Step 5: Calculate the result

Perimeter = 2 × 19 = 38 feet

Perimeter = 38 feet
Final answer:

The perimeter of the room is 38 feet.

Applied rules:

• Relationship solving: Use given relationship to find unknown dimension

• Rectangle formula: P = 2 × (l + w)

5 Fencing problem
Exercise 5
A rectangular garden measures 15 feet long and 10 feet wide.
How many feet of fencing are needed to go around the entire garden?
Definition:

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

🌱
Garden
15ft × 10ft
→
P=2(15+10)
Fence needed
→
50ft
Fencing
Step 1: Identify the dimensions

Length = 15 feet, Width = 10 feet

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Step 3: Substitute the values

Perimeter = 2 × (15 + 10)

Step 4: Calculate inside the parentheses

Perimeter = 2 × 25

Step 5: Complete the multiplication

Perimeter = 50 feet

Fencing needed: 50 feet
Final answer:

50 feet of fencing are needed to go around the garden.

Applied rules:

• Real-world application: Perimeter equals amount of fencing needed

• Rectangle formula: P = 2 × (l + w)

Advanced Rectangle Perimeter Applications and Problem Solving
P = 2(l + w)
Rectangle Perimeter Formula
Key definitions:

Perimeter: The total distance around the outside of a two-dimensional shape

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

Linear measurement: Measurement of distance using units like meters, centimeters, feet, inches

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 perimeter formula P = 2(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: For rectangles, you only need to measure length and width once each.
Tip 4: Check your work by adding all four sides separately to verify the formula result.
Common errors: Mixing units, forgetting to multiply by 2, using area formula instead of perimeter.
Exam preparation: Practice with various units and relationship problems.
Properties to know by heart:

• Rectangle: P = 2 × (length + width)

• Square: P = 4 × side length

• Perimeter is always in linear units

• Opposite sides of rectangle are equal

• Always check for consistent units

Extended Learning Summary:

Advanced Rectangle Perimeter 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 P = 2(l + w) for rectangles
  • Verification: Double-check calculations using alternative methods

Real-world Applications:

  • Fencing: Calculating fence length needed for rectangular yards
  • Trim work: Determining trim needed for rectangular doors, windows, frames
  • Border materials: Calculating edging for rectangular gardens or walkways
  • Picture framing: Measuring frame material needed for rectangular pictures

Advanced Problem Types:

  • Relationship problems with proportional dimensions
  • Missing dimension problems where perimeter is known
  • Unit conversion problems requiring measurement changes
  • Composite problems combining perimeter 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 P = 2(l + w) will give the same result regardless:

  • As long as you identify both different measurements
  • Then add them together and multiply by 2
  • For example: if sides are 5 and 8, P = 2(5 + 8) = 26, same as P = 2(8 + 5) = 26

Just make sure you include both measurements in your calculation!

Question: Can I just add all four sides instead of using the formula P = 2(l + w)?

Answer: Yes, absolutely! Adding all four sides works perfectly:

  • For rectangles: l + w + l + w = 2l + 2w = 2(l + w)
  • Both methods give the same result
  • Adding sides can help verify your formula calculation
  • It's especially helpful when you're learning the concept

The formula is just a shortcut for adding all four sides!

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

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

  • Start with the formula: P = 2(l + w)
  • Divide the perimeter by 2: P/2 = l + w
  • Subtract the known dimension: unknown = P/2 - known dimension
  • For example: if P = 20 and l = 6, then w = 20/2 - 6 = 10 - 6 = 4

This is called a "missing dimension" problem!

Question: Why do we multiply by 2 in the rectangle perimeter formula?

Answer: We multiply by 2 because a rectangle has two pairs of equal sides:

  • A rectangle has 4 sides: 2 lengths and 2 widths
  • So the perimeter is: length + width + length + width
  • This can be rewritten as: 2 × length + 2 × width
  • Factoring out the 2: 2 × (length + width)

The multiplication by 2 accounts for having two of each dimension!

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

Answer: Perimeter and area measure different aspects of a rectangle:

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

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