Solved Exercises on Understanding Perimeter in Grade 4

Master perimeter: calculating perimeters of rectangles, squares, and irregular shapes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle perimeter
Exercise 1
A rectangular garden has a length of 12 meters and a width of 8 meters.
What is the perimeter of the garden?
Definition:

Perimeter: The distance around the outside of a shape

Perimeter method for rectangles:
  1. Identify the length and width of the rectangle
  2. Use the formula: Perimeter = 2 × (length + width)
  3. Or add all four sides: length + width + length + width
  4. Substitute the known values
  5. Calculate the result
📏
Length
12m
+
📐
Width
8m
+
📏
Length
12m
+
📐
Width
8m
=
40m
Perimeter
Step 1: Identify the dimensions

Length = 12 meters, Width = 8 meters

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Step 3: Substitute the values

Perimeter = 2 × (12 + 8)

Step 4: Calculate inside the parentheses

Perimeter = 2 × 20

Step 5: Complete the multiplication

Perimeter = 40 meters

Perimeter = 40 meters
Final answer:

The perimeter of the garden is 40 meters.

Applied rules:

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

• Order of operations: Parentheses first, then multiplication

2 Square perimeter
Exercise 2
A square picture frame has sides that are each 15 centimeters long.
What is the perimeter of the frame?
Definition:

Square: A rectangle with all sides equal in length

⬜
Square
All sides = 15cm
→
15cm
Side 1
+
15cm
Side 2
+
15cm
Side 3
+
15cm
Side 4
=
60cm
Perimeter
Step 1: Identify the side length

All sides of the square = 15 centimeters

Step 2: Apply the square perimeter formula

Perimeter = 4 × side length

Step 3: Substitute the value

Perimeter = 4 × 15

Step 4: Calculate the result

Perimeter = 60 centimeters

Step 5: Verify using rectangle formula

Perimeter = 2 × (15 + 15) = 2 × 30 = 60 cm ✓

Perimeter = 60 centimeters
Final answer:

The perimeter of the square frame is 60 centimeters.

Applied rules:

• Square perimeter: P = 4 × side length

• All sides equal: Each side measures the same in a square

3 Irregular shape perimeter
Exercise 3
A playground has sides measuring 20 feet, 15 feet, 12 feet, and 18 feet.
What is the perimeter of the playground?
Definition:

Irregular polygon: A shape with sides of different lengths

20ft
Side 1
+
15ft
Side 2
+
12ft
Side 3
+
18ft
Side 4
=
65ft
Perimeter
Step 1: List all side lengths

Sides: 20 feet, 15 feet, 12 feet, 18 feet

Step 2: Apply the general perimeter rule

Perimeter = sum of all side lengths

Step 3: Add all sides

Perimeter = 20 + 15 + 12 + 18

Step 4: Calculate the sum

Perimeter = 65 feet

Step 5: Verify the calculation

(20 + 15) + (12 + 18) = 35 + 30 = 65 feet ✓

Perimeter = 65 feet
Final answer:

The perimeter of the playground is 65 feet.

Applied rules:

• General perimeter: Sum of all side lengths

• Irregular shapes: Each side may have a different length

Perimeter Formulas and Rules
P = 2(l + w)
Rectangle Perimeter
Rectangle
P = 2(l + w)
Length + Width × 2
Square
P = 4s
4 × Side Length
Any Polygon
P = Sum of all sides
Add all sides
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°

Square: A rectangle with all four sides equal in length

Side: A straight line segment that forms part of the boundary of a shape

Formula: A mathematical rule expressed using symbols

Perimeter calculation methods:
  1. Rectangle: P = 2 × (length + width)
  2. Square: P = 4 × side length
  3. Regular polygon: P = number of sides × length of one side
  4. Irregular polygon: P = sum of all side lengths
  5. Verification: Double-check by counting or adding separately
Key Property: Perimeter is always measured in linear units (meters, centimeters, feet).
Key Property: Perimeter is the total distance around the outside of a shape.
Detailed Summary:

Understanding Perimeter - Complete Guide

Definitions:

  • Perimeter: Distance around the outside of a shape
  • Rectangle: Four-sided shape with opposite sides equal
  • Square: Rectangle with all sides equal
  • Formula: Mathematical rule for calculations

Core Rules:

  • Rectangle perimeter: P = 2 × (length + width)
  • Square perimeter: P = 4 × side length
  • Any polygon: P = sum of all sides
  • Always use consistent units of measurement

Methods:

  • Identify the shape type
  • Select the appropriate formula
  • Substitute the known values
  • Calculate the result

Examples:

  • Rectangle: 12m × 8m → P = 2(12+8) = 40m
  • Square: 15cm sides → P = 4×15 = 60cm
  • Irregular: 20+15+12+18 = 65ft

Tips & Tricks:

  • Always check that you've included all sides
  • Remember to use the correct units
  • For rectangles, only measure length and width once each
  • Draw a diagram to visualize the shape

Common Mistakes:

  • Forgetting to count all sides
  • Mixing different units of measurement
  • Using area formula instead of perimeter
  • Counting length and width twice in rectangle formula

Memorization Aids:

  • Rectangle: "2 times length plus width"
  • Square: "4 times any side"
  • General: "Add all the sides"
Solution: Exercises 4 to 5
4 Table perimeter problem
Exercise 4
A rectangular table has a length that is twice its width. If the width is 3 feet,
what is the perimeter of the table?
Definition:

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

3ft
Width
→
6ft
Length
2×Width
→
P=2(6+3)
Formula
→
18ft
Perimeter
Step 1: Identify the relationship

Length = 2 × width

Step 2: Find the length

Length = 2 × 3 = 6 feet

Step 3: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Step 4: Substitute the values

Perimeter = 2 × (6 + 3)

Step 5: Calculate the result

Perimeter = 2 × 9 = 18 feet

Perimeter = 18 feet
Final answer:

The perimeter of the table is 18 feet.

Applied rules:

• Relationship solving: Use given relationship to find unknown dimension

• Rectangle formula: P = 2 × (l + w)

5 Door frame problem
Exercise 5
A door frame is 80 inches tall and 36 inches wide. If trim is added around
the entire perimeter, how many inches of trim are needed?
Definition:

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

🚪
Door Frame
80in × 36in
→
P=2(80+36)
Perimeter formula
→
232in
Trim needed
Step 1: Identify the dimensions

Height (length) = 80 inches, Width = 36 inches

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (height + width)

Step 3: Substitute the values

Perimeter = 2 × (80 + 36)

Step 4: Calculate inside the parentheses

Perimeter = 2 × 116

Step 5: Complete the multiplication

Perimeter = 232 inches

Trim needed: 232 inches
Final answer:

232 inches of trim are needed for the door frame.

Applied rules:

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

• Rectangle formula: P = 2 × (l + w)

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

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

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 shape and its dimensions
  3. Select: Choose the appropriate perimeter formula
  4. Substitute: Replace variables with known values
  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 shape 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 sides separately to verify the formula result.
Common errors: Mixing units, forgetting to multiply by 2 for rectangles, counting sides twice.
Exam preparation: Practice with various shapes and unit conversions.
Properties to know by heart:

• Rectangle: P = 2 × (length + width)

• Square: P = 4 × side length

• Any polygon: P = sum of all side lengths

• Perimeter is always in linear units

• Always check for consistent units

Extended Learning Summary:

Advanced Perimeter Concepts

Problem-Solving Strategies:

  • Diagram drawing: Visualize the problem with a simple sketch
  • Unit consistency: Ensure all measurements use the same units
  • Formula selection: Choose the right formula based on the shape
  • Verification: Double-check calculations using alternative methods

Real-world Applications:

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

Advanced Problem Types:

  • Relationship problems with proportional dimensions
  • Compound shapes made of multiple rectangles
  • Missing dimension problems where perimeter is known
  • Unit conversion problems requiring measurement changes

Questions & Answers

Question: I sometimes forget which formula to use for perimeter. How can I remember them?

Answer: Here are some helpful memory aids:

  • Rectangle: "2 times length plus width" → P = 2(l + w)
  • Square: "4 times any side" → P = 4s
  • General: "Add all the sides together"
  • Remember: Perimeter is the distance around the outside

Practice with different shapes to reinforce the patterns!

Question: What if a rectangle has sides labeled differently? How do I know which is length and which is width?

Answer: It doesn't matter which side you call length and which you call width! The formula P = 2(l + w) will work the same way:

  • 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

Just make sure you include both measurements in your calculation!

Question: Can I just add all the sides instead of using formulas?

Answer: Yes, you absolutely can! Adding all sides works for any shape:

  • For rectangles: l + w + l + w = 2l + 2w = 2(l + w)
  • For squares: s + s + s + s = 4s
  • For irregular shapes: just add all the different side lengths

Formulas are shortcuts, but adding sides always works and can help verify your answer!

Question: What if the shape has more than 4 sides? How do I find the perimeter?

Answer: For any polygon (shape with straight sides), the perimeter is always the sum of all side lengths:

  • Triangle: add all 3 sides
  • Pentagon: add all 5 sides
  • Hexagon: add all 6 sides
  • Any polygon: add all the sides

The definition is the same: perimeter is the distance around the outside!

Question: How is perimeter different from area?

Answer: Perimeter and area are different measurements:

  • Perimeter: Distance around the outside (linear units like meters, feet)
  • Area: Space inside the shape (square units like square meters, square feet)
  • Think of perimeter as the fence around a yard, area as the grass inside

They measure different aspects of a shape!