Intermediate Perimeter Word Problems for Grade 4

Master intermediate perimeter calculations: rectangles, squares, triangles, and complex shapes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle with missing dimension
Exercise 1
A rectangular room has a width of 7 meters and a perimeter of 26 meters. What is the length of the room?
Definition:

Perimeter Formula Rearrangement: If P = 2(l + w), then l = (P/2) - w

Missing dimension method:
  1. Start with the perimeter formula: P = 2(l + w)
  2. Substitute the known values
  3. Rearrange the equation to solve for the unknown
  4. Calculate the missing dimension
Given
P = 26m, w = 7m
Formula
l = (P/2) - w
Result
l = 6m
Step 1: Write the perimeter formula

Perimeter = 2 × (length + width)

26 = 2 × (length + 7)

Step 2: Divide both sides by 2

26 ÷ 2 = length + 7

13 = length + 7

Step 3: Subtract 7 from both sides

13 - 7 = length

6 = length

Step 4: Verify the answer

Perimeter = 2 × (6 + 7) = 2 × 13 = 26 ✓

Step 5: State the answer

The length of the room is 6 meters

The length of the room is 6 meters
Final answer:

The length of the rectangular room is 6 meters

Applied rules:

• Perimeter Formula: P = 2(length + width)

• Algebraic Manipulation: Rearrange to solve for unknown

• Verification: Always check your answer

2 Multiple shapes comparison
Exercise 2
A square garden has sides of 9 feet. A rectangular garden has a length of 12 feet and a width of 6 feet. Which garden has a larger perimeter? By how much?
Definition:

Comparison Problems: Calculate each shape's perimeter and find the difference.

Square
P = 4 × 9 = 36ft
Rectangle
P = 2(12+6) = 36ft
Difference
0ft
Step 1: Calculate the square's perimeter

Square: P = 4 × side = 4 × 9 = 36 feet

Step 2: Calculate the rectangle's perimeter

Rectangle: P = 2 × (length + width) = 2 × (12 + 6) = 2 × 18 = 36 feet

Step 3: Compare the perimeters

Square perimeter: 36 feet, Rectangle perimeter: 36 feet

Step 4: Find the difference

Difference = 36 - 36 = 0 feet

Step 5: State the conclusion

Both gardens have the same perimeter of 36 feet

Both gardens have the same perimeter of 36 feet
Final answer:

Both gardens have the same perimeter of 36 feet. There is no difference between them.

Applied rules:

• Square Perimeter: P = 4 × side

• Rectangle Perimeter: P = 2(length + width)

• Comparison: Subtract smaller from larger to find difference

3 Triangle with equal sides
Exercise 3
An equilateral triangle has a perimeter of 24 centimeters. What is the length of each side?
Definition:

Equilateral Triangle: A triangle with all three sides equal. P = 3 × side

Given
P = 24cm
Formula
s = P ÷ 3
Result
s = 8cm
Step 1: Identify the triangle type

Equilateral triangle: all three sides are equal

Step 2: Write the perimeter formula

Perimeter = side + side + side = 3 × side

24 = 3 × side

Step 3: Solve for the side length

Side = 24 ÷ 3 = 8 centimeters

Step 4: Verify the answer

Check: 8 + 8 + 8 = 24 ✓

Step 5: State the answer

Each side of the triangle is 8 centimeters

Each side of the triangle is 8 centimeters
Final answer:

Each side of the equilateral triangle is 8 centimeters long

Applied rules:

• Equilateral Triangle: P = 3 × side

• Division: To find one side, divide perimeter by 3

• Verification: Multiply the side by 3 to check

Intermediate Perimeter Rules and Methods
Rectangle: \(P = 2(l + w)\)
Rectangle Perimeter
Square
\(P = 4s\)
Perimeter of Square
Triangle
\(P = a + b + c\)
Perimeter of Triangle
Equilateral Triangle
\(P = 3s\)
Special Case
Key definitions:

Perimeter: The distance around the outside of a shape

Rectangle: A 4-sided shape with 4 right angles and opposite sides equal

Square: A rectangle with all 4 sides equal

Equilateral Triangle: A triangle with all 3 sides equal

Intermediate methodology:
  1. Identify the problem type: Find missing dimension, compare shapes, or find equal parts
  2. Set up the equation: Use the appropriate perimeter formula
  3. Solve algebraically: Rearrange the formula if needed
  4. Verify the solution: Check if your answer makes sense
  5. State the answer: Include proper units
Tip 1: For missing dimensions, rearrange the perimeter formula.
Tip 2: For comparison problems, calculate each perimeter separately first.
Tip 3: Equilateral triangles: divide perimeter by 3 to find each side.
Tip 4: Always verify your answer by substituting back into the original formula.
Common errors: Forgetting to divide by 2 in rectangle problems, mixing up formulas, omitting units.
Key memorization: Rectangle: P=2(l+w), Square: P=4s, Equilateral Triangle: P=3s.
Formulas to remember:

• Rectangle: \(P = 2 \times (length + width)\)

• Square: \(P = 4 \times side\)

• Triangle: \(P = side_1 + side_2 + side_3\)

• Equilateral Triangle: \(P = 3 \times side\)

• To find missing dimension: \(length = (P \div 2) - width\)

Solution: Exercises 4 to 5
4 Composite shape perimeter
Exercise 4
A shape is made by placing a square with sides of 4 cm on top of a rectangle with dimensions 6 cm by 4 cm. What is the perimeter of the combined shape?
Definition:

Composite Shape Perimeter: Only count the outer edges, not internal connections.

Dimensions
Square: 4×4cm, Rectangle: 6×4cm
External sides
6+4+6+4+4+4
Result
P = 28cm
Step 1: Visualize the composite shape

Square (4×4cm) sits on top of rectangle (6×4cm), sharing one 4cm side

Step 2: Identify all external sides

Bottom rectangle: bottom (6cm), left (4cm), right (4cm)

Square: left (4cm), right (4cm), top (4cm)

Shared side (4cm) is internal, not part of perimeter

Step 3: Add all external sides

Perimeter = 6 + 4 + 4 + 4 + 4 + 4 = 28 cm

Step 4: Verify by counting

Bottom: 6cm, Left sides: 4+4=8cm, Right sides: 4+4=8cm, Top: 4cm

Total: 6 + 8 + 8 + 4 = 26cm

Wait, let's recount: 6(bottom) + 4(left rect) + 4(top left) + 4(top) + 4(top right) + 4(right rect) = 26cm

Step 5: Correct calculation

Actually: 6 + 4 + 4 + 4 + 4 + 4 = 26 cm

The perimeter of the combined shape is 26 cm
Final answer:

The perimeter of the combined shape is 26 centimeters

Applied rules:

• Composite Perimeter: Only count outer edges

• Shared Sides: Internal connections don't contribute to perimeter

• Visualization: Draw the shape to identify external edges

5 Real-world application
Exercise 5
A farmer wants to build a fence around a rectangular field. The field is 45 meters long and 28 meters wide. If fencing costs $3 per meter, how much will it cost to fence the entire field?
Definition:

Cost Application: First find the perimeter, then multiply by the cost per unit length.

Given
Length = 45m, Width = 28m, Cost = $3/m
Steps
P = 2(l+w), Cost = P × rate
Result
$438
Step 1: Find the perimeter of the field

Perimeter = 2 × (length + width)

Perimeter = 2 × (45 + 28) = 2 × 73 = 146 meters

Step 2: Calculate the total cost

Cost = Perimeter × Cost per meter

Cost = 146 × $3 = $438

Step 3: Verify the calculations

Perimeter check: 2 × (45 + 28) = 2 × 73 = 146 ✓

Cost check: 146 × 3 = 438 ✓

Step 4: State the final answer

The total cost to fence the field is $438

The total cost to fence the field is $438
Final answer:

The farmer will need $438 to fence the entire rectangular field

Applied rules:

• Rectangle Perimeter: P = 2(length + width)

• Cost Calculation: Total Cost = Perimeter × Rate per unit

• Multi-step Problems: Solve one step at a time

Complete Perimeter Summary for Grade 4
Rectangle: \(P = 2(l + w)\)
Intermediate Perimeter Concepts
Key definitions:

Perimeter: The distance around the outside of a shape

Rectangle: A 4-sided shape with 4 right angles and opposite sides equal

Square: A rectangle with all 4 sides equal

Equilateral Triangle: A triangle with all 3 sides equal

Composite Shape: A shape made by combining two or more basic shapes

Intermediate methodology:
  1. Identify the shape: Determine if it's rectangle, square, triangle, or composite
  2. Find known values: Locate given measurements
  3. Select appropriate formula: Use the correct perimeter formula
  4. Solve systematically: Apply algebra if needed for missing values
  5. Verify and state: Check your work and provide complete answer
Tip 1: For missing dimensions, rearrange the perimeter formula: l = (P/2) - w
Tip 2: In composite shapes, only count the outer edges, not internal connections.
Tip 3: For equilateral triangles, divide the perimeter by 3 to get each side.
Tip 4: Always verify your answer by substituting back into the original formula.
Common errors: Forgetting to divide by 2 in rectangle problems, counting internal sides in composite shapes, mixing up formulas.
Advanced strategies: Break complex problems into smaller parts, use algebra for missing values, draw diagrams to visualize.
Essential formulas for grade 4:

• Rectangle: \(P = 2 \times (length + width)\)

• Square: \(P = 4 \times side\)

• Triangle: \(P = side_1 + side_2 + side_3\)

• Equilateral Triangle: \(P = 3 \times side\)

• Missing Length: \(length = (perimeter \div 2) - width\)

• Cost Problems: \(total\_cost = perimeter \times cost\_per\_unit\)

Perimeter Applications in Daily Life
Perimeter Uses
Perimeter calculations in real life:
Fencing: rectangular yards, square gardens
Construction: room measurements, baseboards
Crafts: framing pictures, making borders

Applications: Understanding perimeter is essential for practical tasks.

  • Fencing: Need to know perimeter to purchase the correct amount of material
  • Construction: Baseboard installation requires perimeter measurements
  • Crafts: Picture framing and border creation
  • Gardening: Planning pathways and garden beds

Questions & Answers

Question: How do I find a missing dimension if I know the perimeter and one side of a rectangle?

Answer: You can rearrange the perimeter formula to solve for the missing dimension:

  • Start with: P = 2 × (length + width)
  • If you know P and width, solve for length: length = (P ÷ 2) - width
  • If you know P and length, solve for width: width = (P ÷ 2) - length

Example: If perimeter is 20 and width is 4, then length = (20 ÷ 2) - 4 = 10 - 4 = 6.

Always verify: 2 × (6 + 4) = 2 × 10 = 20 ✓

Question: My child gets confused when calculating the perimeter of composite shapes. How can I help them?

Answer: Here's a strategy to help your child understand composite shapes:

  1. Draw and label: Have your child sketch the composite shape and label all known dimensions
  2. Identify the rule: Explain that only the outer edges count toward the perimeter
  3. Trace the path: Have them trace the outer boundary with their finger to visualize which sides to include
  4. Mark internal sides: Show them to cross out any sides that are inside the shape (not part of the outer boundary)
  5. Count carefully: Add only the sides that form the outside edge

Practice with simple examples first, then gradually increase complexity. Drawing helps visualize which sides contribute to the perimeter.

Question: What's an equilateral triangle, and how do I find the side length if I know the perimeter?

Answer: An equilateral triangle is a special triangle where all three sides are exactly the same length. Because all sides are equal:

  • Perimeter = side + side + side = 3 × side
  • To find one side: side = perimeter ÷ 3

Example: If the perimeter is 15 cm, then each side = 15 ÷ 3 = 5 cm.

Check: 5 + 5 + 5 = 15 ✓

This works because all three sides are equal in an equilateral triangle!