Solved Exercises on Perimeter Word Problems in Grade 3

Master perimeter calculations: rectangles, squares, triangles, and complex 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 total distance around the edge of a shape. For a rectangle: \(P = 2 \times (length + width)\)

Perimeter calculation method:
  1. Add all sides of the rectangle
  2. Or use the formula: \(P = 2 \times (length + width)\)
  3. Always include units in your final answer
Given
Length = 12m, Width = 8m
Formula
P = 2(l + w)
Result
P = 40m
Step 1: Identify given values

Length = 12 meters, Width = 8 meters

Step 2: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (12 + 8)

Step 3: Calculate the sum inside parentheses

12 + 8 = 20

Step 4: Multiply by 2

2 × 20 = 40

Step 5: Add units

Perimeter = 40 meters

The perimeter of the garden is 40 meters
Final answer:

The perimeter of the rectangular garden is 40 meters

Applied rules:

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

• Order of Operations: Calculate parentheses first

• Units: Always include units in final answer

2 Square perimeter
Exercise 2
A square playground has sides of 15 feet each. What is the perimeter of the playground?
Definition:

Square Perimeter: Since all sides are equal, \(P = 4 \times side\)

Given
Side = 15 ft
Formula
P = 4s
Result
P = 60 ft
Step 1: Identify the side length

All sides of a square are equal: Side = 15 feet

Step 2: Apply the square perimeter formula

Perimeter = 4 × side length

Perimeter = 4 × 15

Step 3: Calculate the multiplication

4 × 15 = 60

Step 4: Add units

Perimeter = 60 feet

The perimeter of the playground is 60 feet
Final answer:

The perimeter of the square playground is 60 feet

Applied rules:

• Square Perimeter: P = 4 × side length

• Equal Sides: All four sides of a square are equal

• Units: Include units in final answer

3 Triangle perimeter
Exercise 3
A triangular fence has sides measuring 10m, 12m, and 8m. What is the perimeter of the fence?
Definition:

Triangle Perimeter: Sum of all three sides, \(P = a + b + c\)

Given
Sides: 10m, 12m, 8m
Formula
P = a + b + c
Result
P = 30m
Step 1: Identify all three sides

Side 1 = 10m, Side 2 = 12m, Side 3 = 8m

Step 2: Apply the triangle perimeter formula

Perimeter = Side 1 + Side 2 + Side 3

Perimeter = 10 + 12 + 8

Step 3: Calculate the sum

10 + 12 + 8 = 30

Step 4: Add units

Perimeter = 30 meters

The perimeter of the triangular fence is 30 meters
Final answer:

The perimeter of the triangular fence is 30 meters

Applied rules:

• Triangle Perimeter: P = sum of all three sides

• Addition: Add all measurements together

• Units: Include units in final answer

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
Any Shape
\(P = \text{sum of all sides}\)
General Rule
Key definitions:

Perimeter: The distance around the outside 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°

Triangle: Three-sided polygon

Complete methodology:
  1. Read the problem carefully: Identify the shape and given measurements
  2. Choose the correct formula: Rectangle, square, or triangle
  3. Substitute values: Replace variables with given numbers
  4. Calculate: Follow order of operations
  5. Check units: Include proper units in final answer
Tip 1: Remember: perimeter means "around the outside".
Tip 2: Draw a picture to visualize the shape and its dimensions.
Tip 3: Always write down the formula before substituting numbers.
Tip 4: Check if your answer makes sense compared to the given dimensions.
Common errors: Forgetting to multiply by 2 for rectangles, adding instead of multiplying for squares, omitting units.
Key memorization: Rectangle P=2(l+w), Square P=4s, Triangle P=a+b+c.
Formulas to know by heart:

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

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

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

• Any Polygon: \(P = \text{sum of all sides}\)

Solution: Exercises 4 to 5
4 Complex shape perimeter
Exercise 4
A room has the shape of an L. It has sides measuring 6m, 4m, 2m, 4m, 8m, and 2m. What is the perimeter of the room?
Definition:

Complex Shape Perimeter: Add up the lengths of all sides, regardless of the shape's complexity

Given
Sides: 6, 4, 2, 4, 8, 2 m
Formula
P = sum of all sides
Result
P = 26m
Step 1: List all side measurements

6m, 4m, 2m, 4m, 8m, 2m

Step 2: Apply the general perimeter rule

Perimeter = sum of all side lengths

Step 3: Add all measurements together

6 + 4 + 2 + 4 + 8 + 2 = 26

Step 4: Add units

Perimeter = 26 meters

The perimeter of the L-shaped room is 26 meters
Final answer:

The perimeter of the L-shaped room is 26 meters

Applied rules:

• General Rule: Perimeter = sum of all sides

• Complex Shapes: Count every side exactly once

• Addition: Carefully add all measurements

5 Real-world application
Exercise 5
Sarah wants to put a fence around her rectangular vegetable garden. The garden is 18 feet long and 10 feet wide. How many feet of fencing does she need?
Definition:

Real-world Perimeter: The amount of material needed to go around a shape

Given
Length = 18ft, Width = 10ft
Formula
P = 2(l + w)
Result
P = 56ft
Step 1: Identify the shape and dimensions

Rectangular garden: Length = 18 feet, Width = 10 feet

Step 2: Understand the real-world context

Fencing needed = perimeter of the garden

Step 3: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (18 + 10)

Step 4: Calculate the sum in parentheses

18 + 10 = 28

Step 5: Multiply by 2

2 × 28 = 56

Step 6: State the answer with units

Sarah needs 56 feet of fencing

Sarah needs 56 feet of fencing
Final answer:

Sarah needs 56 feet of fencing to surround her garden

Applied rules:

• Real-world Context: Perimeter represents the amount of material needed

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

• Units: Answer must match the unit of measurement given

Perimeter Laws, Methods, Rules, and Definitions
Rectangle: \(P = 2(l + w)\)
Perimeter Calculation
Key definitions:

Perimeter: The total distance around the boundary of a 2D shape

Rectangle: A quadrilateral with four right angles and opposite sides equal

Square: A rectangle with all four sides equal

Triangle: A polygon with three sides and three angles

Complete methodology:
  1. Identify the shape: Determine whether it's rectangle, square, triangle, or complex polygon
  2. Count the sides: Make sure you have measurements for all sides
  3. Select the appropriate formula: Use the correct perimeter formula for the shape
  4. Substitute values: Replace variables with the given measurements
  5. Calculate: Perform the arithmetic operations carefully
  6. Check your work: Verify your answer makes sense
Tip 1: Draw the shape and label all known sides to visualize the problem.
Tip 2: For rectangles, remember: perimeter = twice the length plus twice the width.
Tip 3: Always double-check that you've included all sides in your calculation.
Tip 4: Estimate your answer first to check if your calculated result is reasonable.
Common errors: Forgetting to multiply by 2 in rectangle formula, counting some sides twice, omitting units.
Exam preparation: Memorize formulas, practice word problems, focus on unit consistency.
Formulas to know by heart:

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

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

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

• Any Polygon: \(P = \text{sum of all sides}\)

• Units: Always include the unit of measurement in your answer

Visual Perimeter Comparison: Different Shapes
Comparing Perimeters
Let's compare perimeters of different shapes with the same area:
Square with side 4 units (Area = 16 sq units)
Rectangle with length 8, width 2 (Area = 16 sq units)
Right triangle with legs 6 and 5.33 (Area ≈ 16 sq units)

Analysis: Even with the same area, shapes can have different perimeters.

  • Square: P = 4×4 = 16 units (most efficient perimeter for given area)
  • Rectangle: P = 2(8+2) = 20 units
  • Right Triangle: P ≈ 6 + 5.33 + 7.81 ≈ 19.14 units

Questions & Answers

Question: I don't understand why we multiply by 2 for rectangles but by 4 for squares. Can you explain the difference?

Answer: Great question! The reason is based on the properties of rectangles and squares:

  • A rectangle has two pairs of equal sides: length and width. So the perimeter is length + width + length + width = 2×length + 2×width = 2×(length + width)
  • A square has four equal sides. So the perimeter is side + side + side + side = 4×side

Actually, if you apply the rectangle formula to a square where length = width = s, you get: P = 2×(s + s) = 2×(2s) = 4s. Both formulas work, but the square formula is simpler!

Example: For a square with sides of 5cm: Rectangle formula gives P = 2×(5+5) = 2×10 = 20cm, and square formula gives P = 4×5 = 20cm. Same result!

Question: When solving word problems, how do I know if I need to find perimeter or area?

Answer: Look for clue words in the problem:

  • Perimeter clues: "fence around", "border", "around the outside", "distance around", "trim", "frame", "walk around"
  • Area clues: "covering", "carpet", "paint", "tiles", "grass", "inside", "surface"

Examples:

  • "How much fence is needed?" → Perimeter
  • "How much carpet is needed?" → Area
  • "What is the distance around the field?" → Perimeter
  • "How much paint is needed to cover the wall?" → Area

Remember: Perimeter is about the edge/boundary, Area is about the space inside.

Question: My child sometimes forgets to include units in their answers. Why is this important?

Answer: Including units is crucial for several reasons:

  • Meaning: Numbers without units have no real-world meaning. 10 could mean 10 inches, 10 feet, or 10 miles!
  • Communication: Units tell others exactly what you measured and how big it is
  • Problem-solving: Units help catch errors. If you expect an answer in meters but get centimeters, you know something's wrong
  • Real-world applications: In construction, engineering, and science, units are critical for safety and accuracy

Example: If the perimeter is 24, that's meaningless. But 24 feet tells us exactly the distance around the shape. Always encourage your child to end their answer with the appropriate unit!