Solved Exercises on Perimeter Word Problems in Grade 3

Master perimeter calculations: rectangles, squares, polygons with real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle Perimeter
Exercise 1
Sarah has a rectangular garden that is 8 feet long and 5 feet wide. She wants to put a fence around the entire garden. How much fencing does she need?
Definition:

Perimeter: The distance around the outside of a shape

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

Perimeter calculation method:
  1. Identify the length and width of the rectangle
  2. Add the length and width together
  3. Multiply the sum by 2
Given
Length = 8 ft, Width = 5 ft
Formula
P = 2(l + w)
Result
26 ft
Step 1: Identify given information

Length = 8 feet, Width = 5 feet

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

P = 2 × (8 + 5)

Step 3: Calculate inside the parentheses

P = 2 × (13)

Step 4: Multiply by 2

P = 2 × 13 = 26 feet

Sarah needs 26 feet of fencing
Final answer:

Sarah needs 26 feet of fencing to surround her garden

Applied rules:

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

• Order of Operations: Solve parentheses first, then multiply

• Units: Always include units in your final answer

2 Square Perimeter
Exercise 2
A square picture frame has sides that are 12 inches long. What is the total length of the wooden trim needed to make the frame?
Definition:

Square Perimeter: \(P = 4 \times side\)

All sides of a square are equal in length

Given
Side = 12 in
Formula
P = 4s
Result
48 in
Step 1: Identify the side length

Each side of the square = 12 inches

Step 2: Apply the square perimeter formula

Perimeter = 4 × side length

P = 4 × 12

Step 3: Calculate the perimeter

P = 48 inches

The wooden trim needed is 48 inches long
Final answer:

The wooden trim needed is 48 inches long

Applied rules:

• Square Perimeter Formula: P = 4 × side length

• Equal Sides: All 4 sides of a square are the same length

• Multiplication: 4 × side length gives the total perimeter

3 Polygon Perimeter
Exercise 3
A pentagon-shaped playground has sides measuring 10 feet, 12 feet, 8 feet, 15 feet, and 10 feet. What is the total distance around the playground?
Definition:

Polygon Perimeter: Sum of all side lengths

Perimeter = side₁ + side₂ + side₃ + ...

Given
Sides: 10, 12, 8, 15, 10 ft
Formula
P = sum of sides
Result
55 ft
Step 1: List all side lengths

Side 1 = 10 feet, Side 2 = 12 feet, Side 3 = 8 feet, Side 4 = 15 feet, Side 5 = 10 feet

Step 2: Add all side lengths together

Perimeter = 10 + 12 + 8 + 15 + 10

Step 3: Calculate the sum

10 + 12 = 22

22 + 8 = 30

30 + 15 = 45

45 + 10 = 55

Step 4: State the final answer

Perimeter = 55 feet

The distance around the playground is 55 feet
Final answer:

The distance around the playground is 55 feet

Applied rules:

• Polygon Perimeter: Add all side lengths

• Order of Addition: Add sides in any order

• Units: Keep consistent units throughout

Rules and methods, laws,...
\(P = 2(l + w)\)
Rectangle Perimeter
Square
\(P = 4s\)
Perimeter of Square
Polygon
\(P = s_1 + s_2 + s_3 + ...\)
Perimeter of Polygon
Triangle
\(P = a + b + c\)
Perimeter of Triangle
Key definitions:

Perimeter: The total distance around the outside of a shape

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

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

Complete methodology:
  1. Read the problem carefully: Identify what shape is involved
  2. Find the dimensions: Note all side lengths mentioned
  3. Choose the formula: Rectangle, square, or polygon formula
  4. Calculate: Substitute values and solve
  5. Check: Ensure answer makes sense in context
Tip 1: Always write down the formula before substituting numbers.
Tip 2: Draw a picture to visualize the problem.
Tip 3: Don't forget to include units in your final answer.
Tip 4: Check if your answer is reasonable compared to the given measurements.
Common errors: Adding wrong dimensions, forgetting to multiply by 2 for rectangles, omitting units.
Memory trick: "Perimeter" starts with "P" just like "Path" - it's the path around the shape.
Formulas to know by heart:

• Rectangle: P = 2(length + width)

• Square: P = 4 × side

• Triangle: P = side₁ + side₂ + side₃

• Any polygon: P = sum of all sides

Solution: Exercises 4 to 5
4 Mixed Shapes
Exercise 4
Tom is building a sandbox with one side that is 6 feet long, another side that is 6 feet long, a third side that is 4 feet long, and a fourth side that is 4 feet long. What is the perimeter of the sandbox?
Definition:

Rectangle: A four-sided shape with opposite sides equal

This shape is a rectangle with length = 6 ft and width = 4 ft

Given
Two sides = 6 ft, Two sides = 4 ft
Formula
P = 2(l + w)
Result
20 ft
Step 1: Identify the shape and dimensions

This is a rectangle with length = 6 feet and width = 4 feet

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

P = 2 × (6 + 4)

Step 3: Calculate inside the parentheses

P = 2 × (10)

Step 4: Multiply by 2

P = 2 × 10 = 20 feet

The perimeter of the sandbox is 20 feet
Final answer:

The perimeter of the sandbox is 20 feet

Applied rules:

• Shape Recognition: Two pairs of equal sides indicate a rectangle

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

• Substitution: Replace variables with given values

5 Real-world Application
Exercise 5
A farmer wants to build a fence around a triangular field. One side of the field measures 25 meters, another side measures 30 meters, and the third side measures 20 meters. How much fencing does the farmer need?
Definition:

Triangle: A three-sided polygon

Triangle Perimeter: P = side₁ + side₂ + side₃

Given
Sides: 25m, 30m, 20m
Formula
P = a + b + c
Result
75m
Step 1: Identify the shape and side lengths

This is a triangle with sides of 25 meters, 30 meters, and 20 meters

Step 2: Apply the triangle perimeter formula

Perimeter = side₁ + side₂ + side₃

P = 25 + 30 + 20

Step 3: Add the side lengths

25 + 30 = 55

55 + 20 = 75 meters

Step 4: State the final answer

The farmer needs 75 meters of fencing

The farmer needs 75 meters of fencing
Final answer:

The farmer needs 75 meters of fencing to surround the triangular field

Applied rules:

• Triangle Perimeter: Add all three side lengths

• Real-world Context: Fencing represents the perimeter

• Units: Answer should match the unit of measurement given

Key Perimeter Concepts and Formulas
\(P = 2(l + w)\)
Rectangle Perimeter
Key definitions:

Perimeter: The total distance around the edge of a shape

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

Square: Four-sided shape with all sides equal and all angles 90°

Polygon: Many-sided shape with straight sides

Complete methodology:
  1. Identify the shape: Determine if it's a rectangle, square, triangle, or other polygon
  2. Measure or identify dimensions: Note all relevant side lengths
  3. Select the correct formula: Rectangle, square, triangle, or polygon formula
  4. Substitute values: Put the known measurements into the formula
  5. Calculate: Perform the arithmetic operations
  6. State the answer: Include the correct units
Tip 1: Count the number of sides to determine the type of polygon.
Tip 2: For rectangles, you only need length and width since opposite sides are equal.
Tip 3: For squares, measure one side and multiply by 4.
Tip 4: Always check that your answer is larger than any individual side.
Common errors: Forgetting to double the sum for rectangles, mixing up length and width, omitting units.
Real-world applications: Fencing, framing, borders, walking paths, and construction projects.
Essential formulas:

• Rectangle: P = 2(length + width)

• Square: P = 4 × side length

• Triangle: P = side₁ + side₂ + side₃

• Any polygon: P = sum of all side lengths

• Regular polygon: P = number of sides × side length

Visualizing Perimeter Relationships
Perimeter Comparison
Compare perimeters of different shapes with the same area:
Square: 4×4 = 16 sq units → Perimeter = 16 units
Rectangle: 2×8 = 16 sq units → Perimeter = 20 units
Rectangle: 1×16 = 16 sq units → Perimeter = 34 units

Key Insight: Shapes with the same area can have different perimeters. Squares generally have the smallest perimeter for a given area.

  • Square: Most efficient use of perimeter for area
  • Longer rectangles: Higher perimeter for same area
  • Application: Minimizing material costs in construction

Questions & Answers

Question: I don't understand why we multiply by 2 when finding the perimeter of a rectangle. Why can't we just add the length and width once?

Answer: Great question! When we find the perimeter of a rectangle, we need to measure ALL the way around the shape. A rectangle has 4 sides:

  • One length along the top
  • One width along the right side
  • One length along the bottom
  • One width along the left side

So we have TWO lengths and TWO widths. That's why the formula is: P = length + width + length + width = 2×length + 2×width = 2(length + width)

Think of walking around a rectangular garden - you'd walk the length twice (top and bottom) and the width twice (left and right).

Question: How do I know if a shape is a square or a rectangle when solving perimeter problems?

Answer: Look for these clues in the problem:

  • Square: The problem will say "square," or it will tell you that "all sides are equal," or give you only one measurement for side length
  • Rectangle: The problem might say "rectangle," or it will give you two different measurements (length and width), or specify that opposite sides are equal

Example of square: "A square has sides of 5 cm" → All sides are 5 cm

Example of rectangle: "A rectangle is 8 cm long and 4 cm wide" → Two sides are 8 cm, two sides are 4 cm

If you see the word "square," you know all sides are equal!

Question: What's the easiest way to help students remember the different perimeter formulas?

Answer: Here are effective memory strategies for perimeter formulas:

  • Rectangle: "Two Lengths, Two Widths" → P = 2(l + w)
  • Square: "Four Equal Sides" → P = 4s
  • Any Shape: "Add All Sides" → P = sum of all sides

Visual cues help too:

  • Draw arrows going around the shape to show you're measuring the whole boundary
  • Use hands-on activities like string or rulers to physically measure perimeters
  • Create a "formula sheet" with drawings next to each formula

Remember: Perimeter is always about going around the OUTSIDE of the shape!

Question: I sometimes get confused about when to add vs. multiply in perimeter problems. How can I tell the difference?

Answer: The key is to understand what each operation means in the context of perimeter:

  • Addition: We add when we're combining different parts. In polygons, we add all the different side lengths together.
  • Multiplication: We multiply when we have equal groups. For rectangles, we multiply by 2 because there are 2 lengths and 2 widths of equal measure.

Think of it this way:

  • Irregular shapes: Add each unique side (addition)
  • Rectangles: Add length + width, then multiply by 2 (both addition and multiplication)
  • Squares: Multiply one side by 4 (multiplication only)

Always read the problem carefully to understand the shape and its properties!

Question: How can I help my child practice perimeter in real-life situations?

Answer: Real-world applications make perimeter meaningful! Try these activities:

  • Garden planning: Measure the perimeter of garden beds to know how much fencing you need
  • Rug placement: Measure room perimeters to determine where to place rugs
  • Picture frames: Calculate the perimeter of pictures to buy the right frame
  • Walking paths: Walk around your yard or park and estimate the perimeter
  • Furniture arrangement: Measure room perimeters when arranging furniture

These activities connect math to practical life skills while reinforcing the concept that perimeter measures the distance around something.

Make it fun by turning it into a measuring scavenger hunt around your home!