Solved Exercises on Real-World Area and Perimeter Problems in Grade 3

Master area and perimeter through real-world applications: rectangles, squares, conversions, and practical scenarios with these 5 detailed exercises.

Key Concepts and Formulas
Rectangle: Area = Length × Width
Area of Rectangle
Rectangle: Perimeter = 2 × (Length + Width)
Perimeter of Rectangle
Square: Area = Side × Side
Area of Square
Square: Perimeter = 4 × Side
Perimeter of Square
Key definitions:

Area: The amount of space inside a shape, measured in square units (cm², m², etc.)

Perimeter: The distance around the outside of a shape, measured in linear units (cm, m, etc.)

Unit Conversion: Changing measurements from one unit to another (e.g., cm to m)

Problem-solving method:
  1. Read carefully: Identify what is being asked (area or perimeter)
  2. Identify measurements: Find the length and width of the shape
  3. Choose formula: Use the correct area or perimeter formula
  4. Calculate: Substitute values and perform calculations
  5. Check units: Make sure answer has correct units
Tip 1: Remember that area is measured in square units (cm², m²) while perimeter is in linear units (cm, m).
Tip 2: Always write down the formula before substituting numbers.
Tip 3: Draw a picture to visualize the problem when needed.
Tip 4: Check if units need conversion before calculating.
Common conversions: 1 meter = 100 centimeters, 1 kilometer = 1000 meters.
Memorization aid: Area = L×W (like "LW" for "Love"), Perimeter = 2(L+W) (like "2 times Love").
Solution: Exercises 1 to 2
1 Garden Fencing
Exercise 1
Sarah wants to fence her rectangular garden. The garden is 12 meters long and 8 meters wide. How many meters of fencing does she need?
Definition:

Perimeter: The total distance around a shape. For a rectangle: P = 2 × (Length + Width)

Solution method:
  1. Identify the measurements: Length = 12m, Width = 8m
  2. Apply the perimeter formula for a rectangle
  3. Substitute values and calculate
12m
8m
Given
L = 12m, W = 8m
Formula
P = 2(L + W)
Answer
40m
Step 1: Identify the measurements

Length = 12 meters, Width = 8 meters

Step 2: Apply the perimeter formula

Perimeter = 2 × (Length + Width)

P = 2 × (12 + 8)

Step 3: Calculate the sum in parentheses

12 + 8 = 20

Step 4: Multiply by 2

2 × 20 = 40

Step 5: State the answer with units

Sarah needs 40 meters of fencing

Sarah needs 40 meters of fencing
Final answer:

Sarah needs 40 meters of fencing

Applied rules:

• Perimeter formula: P = 2 × (L + W) for rectangles

• Units: Answer in meters since measurements were in meters

• Order of operations: Solve parentheses first, then multiply

2 Carpet Installation
Exercise 2
Tom is buying carpet for his square bedroom. Each side of the room measures 4 meters. How much carpet does he need to cover the floor?
Definition:

Area: The amount of space inside a shape. For a square: Area = Side × Side

4m
Given
Side = 4m
Formula
A = Side²
Answer
16m²
Step 1: Identify the measurement

Each side of the square room = 4 meters

Step 2: Apply the area formula for a square

Area = Side × Side

A = 4 × 4

Step 3: Calculate the area

4 × 4 = 16

Step 4: State the answer with correct units

Tom needs 16 square meters of carpet

Tom needs 16 square meters of carpet
Final answer:

Tom needs 16 square meters of carpet

Applied rules:

• Area formula: A = Side × Side for squares

• Units: Answer in square meters (m²) since measuring area

• Visualization: Square means all sides are equal

Solution: Exercises 3 to 5
3 Tablecloth Design
Exercise 3
Emma is making a tablecloth for a rectangular table. The table is 150 cm long and 90 cm wide. She wants to add a 10 cm border around the entire table. What is the area of the tablecloth she needs to make?
Definition:

Extended dimensions: When adding a border, increase both length and width by twice the border width.

Solution method:
  1. Calculate new dimensions with border included
  2. Apply area formula for rectangle
  3. Substitute values and calculate
150cm
90cm
Step 1: Determine new dimensions with border

New Length = Original Length + 2 × Border Width

New Length = 150 + 2 × 10 = 150 + 20 = 170 cm

New Width = Original Width + 2 × Border Width

New Width = 90 + 2 × 10 = 90 + 20 = 110 cm

Step 2: Apply area formula for rectangle

Area = Length × Width

A = 170 × 110

Step 3: Calculate the area

170 × 110 = 18,700

Step 4: State the answer with correct units

Emma needs a tablecloth with an area of 18,700 square centimeters

Emma needs 18,700 cm² of tablecloth
Final answer:

Emma needs 18,700 square centimeters of tablecloth

Applied rules:

• Border calculation: Add border to both ends of each dimension

• Area formula: A = L × W for rectangles

• Units: Answer in square centimeters (cm²)

4 Playground Fencing
Exercise 4
A school playground is a square with sides measuring 25 meters. They want to install a fence around it, but need to leave a 3-meter gate open. How much fencing material is needed?
Definition:

Adjusted perimeter: Subtract any openings from the total perimeter.

25m
Step 1: Calculate the full perimeter of the square

Perimeter = 4 × Side

P = 4 × 25 = 100 meters

Step 2: Subtract the gate opening from the perimeter

Fencing needed = Full perimeter - Gate width

Fencing needed = 100 - 3 = 97 meters

Step 3: State the final answer

The school needs 97 meters of fencing material

The school needs 97 meters of fencing
Final answer:

The school needs 97 meters of fencing material

Applied rules:

• Perimeter of square: P = 4 × Side

• Adjustment for openings: Subtract from total perimeter

• Real-world application: Gates and doors reduce needed materials

5 Park Renovation
Exercise 5
A city park is rectangular, measuring 150 meters long and 80 meters wide. The city wants to plant flowers along the entire perimeter. If they plant one flower every 2 meters, how many flowers do they need? Also, convert the park's area to square kilometers.
Definition:

Unit conversion: 1 km² = 1,000,000 m². Spacing calculation: Number of items = Perimeter ÷ Spacing interval.

150m
80m
Step 1: Calculate the perimeter of the park

Perimeter = 2 × (Length + Width)

P = 2 × (150 + 80) = 2 × 230 = 460 meters

Step 2: Calculate number of flowers needed

Number of flowers = Perimeter ÷ Spacing

Number of flowers = 460 ÷ 2 = 230 flowers

Step 3: Calculate the area of the park

Area = Length × Width

A = 150 × 80 = 12,000 m²

Step 4: Convert area to square kilometers

1 km² = 1,000,000 m²

Area in km² = 12,000 ÷ 1,000,000 = 0.012 km²

Step 5: State the final answers

The city needs 230 flowers and the park area is 0.012 km²

230 flowers needed, 0.012 km² area
Final answer:

The city needs 230 flowers and the park area is 0.012 square kilometers

Applied rules:

• Perimeter formula: P = 2 × (L + W) for rectangles

• Spacing calculation: Total distance ÷ Interval = Number of items

• Unit conversion: 1 km² = 1,000,000 m²

Comprehensive Guide: Area, Perimeter, and Unit Conversions
Rectangle: A = L × W, P = 2(L + W)
Rectangle Formulas
Square: A = s², P = 4s
Square Formulas
Key definitions:

Area: The amount of surface inside a shape, measured in square units (cm², m², km²).

Perimeter: The total distance around the edge of a shape, measured in linear units (cm, m, km).

Unit conversion: Changing measurements from one unit to another (e.g., cm to m, m² to km²).

Complete methodology:
  1. Understand the problem: Identify if it's asking for area or perimeter
  2. Identify measurements: Find the necessary dimensions
  3. Choose the correct formula: Area or perimeter, rectangle or square
  4. Perform calculations: Substitute values and compute
  5. Convert units if needed: Match the required unit
  6. Check your work: Verify calculations and units
Tip 1: Draw a diagram to visualize the problem, especially for complex scenarios.
Tip 2: When adding borders, remember to add to both sides of each dimension.
Tip 3: Always pay attention to units - area is square units, perimeter is linear units.
Tip 4: For real-world problems, consider practical constraints like gates or borders.
Unit conversions: 1m = 100cm, 1km = 1000m, 1km² = 1,000,000m².
Common errors: Confusing area and perimeter formulas, forgetting unit conversions, miscounting dimensions.
Essential formulas to memorize:

• Rectangle area: A = Length × Width

• Rectangle perimeter: P = 2 × (Length + Width)

• Square area: A = Side × Side

• Square perimeter: P = 4 × Side

• Unit conversion: 1 km² = 1,000,000 m²

Common mistakes to avoid:

• Adding the border width only once instead of twice per dimension

• Using wrong units in the final answer

• Confusing area and perimeter formulas

• Forgetting to subtract openings when calculating fencing requirements

Linear Units Conversion Factor Example
Centimeters to Meters ÷ 100 200 cm = 2 m
Meters to Kilometers ÷ 1000 3000 m = 3 km
Square Centimeters to Square Meters ÷ 10,000 50,000 cm² = 5 m²
Square Meters to Square Kilometers ÷ 1,000,000 2,000,000 m² = 2 km²

Questions & Answers

Question: I get confused between area and perimeter. How do I know which one to calculate when solving a word problem?

Answer: Great question! Here's how to identify whether to calculate area or perimeter:

  • Calculate PERIMETER when: The problem asks about going around, fencing, bordering, or measuring the distance around something.
  • Calculate AREA when: The problem asks about covering, painting, carpeting, or finding how much space is inside something.

Examples:

  • "How much fencing do I need?" → Perimeter
  • "How much carpet covers the floor?" → Area
  • "What's the distance around the garden?" → Perimeter
  • "How much paint is needed for the wall?" → Area

Think of perimeter as walking around the edge of something, and area as covering the entire surface.

Question: When do I need to convert units, and how do I know which conversion to use?

Answer: Unit conversion is needed when:

  • The measurements in the problem are given in different units
  • The answer needs to be in a specific unit different from the given measurements

Common conversions for grade 3:

  • 1 meter = 100 centimeters
  • 1 kilometer = 1000 meters
  • 1 square meter = 10,000 square centimeters

To convert from smaller to larger units, divide. To convert from larger to smaller units, multiply.

Example: If a problem gives measurements in centimeters but asks for an answer in meters, divide by 100.

Question: How do I handle problems where I need to add a border around a shape? It seems tricky.

Answer: Adding borders can seem tricky, but here's the key concept:

When adding a border around a rectangle, you must add the border width to BOTH ends of each dimension. So if you add a 5cm border around a rectangle, you add 5cm to the left AND 5cm to the right (total of 10cm to the length) and 5cm to the top AND 5cm to the bottom (total of 10cm to the width).

Formula: New dimension = Original dimension + 2 × Border width

Example: If a 10cm × 8cm rectangle gets a 2cm border:

  • New length = 10 + 2×2 = 14cm
  • New width = 8 + 2×2 = 12cm

This ensures the border appears equally on all sides of the original shape.