Solved Exercises on Finding Perimeter and Area of Rectangles in Grade 3

Master finding perimeter and area of rectangles: basic formulas, real-life applications, comparison problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic rectangle
Exercise 1
Find the perimeter and area of a rectangle with length 6 units and width 4 units.
Definition:

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

Perimeter: The total distance around the rectangle

Area: The amount of space inside the rectangle

Perimeter and area methods:
  1. Perimeter: Add up all four sides: length + width + length + width
  2. Or use formula: Perimeter = 2 × (length + width)
  3. Area: Multiply length by width: Area = length × width
Given
Length = 6, Width = 4
Perimeter
P = 2(l+w)
Area
A = l×w
Results
P = 20, A = 24
Step 1: Identify the dimensions

Length = 6 units, Width = 4 units

Step 2: Calculate the perimeter

Perimeter = 2 × (length + width)

Perimeter = 2 × (6 + 4)

Perimeter = 2 × 10 = 20 units

Step 3: Calculate the area

Area = length × width

Area = 6 × 4 = 24 square units

Step 4: State both answers

Perimeter = 20 units

Area = 24 square units

Perimeter = 20 units, Area = 24 square units
Final answer:

The perimeter is 20 units and the area is 24 square units

Applied rules:

• Rectangle Property: Opposite sides are equal

• Perimeter Formula: P = 2(l+w) where l is length and w is width

• Area Formula: A = l×w

• Unit Distinction: Perimeter uses linear units, area uses square units

2 Square example
Exercise 2
Find the perimeter and area of a square with side length 5 units.
Definition:

Square: A special rectangle where all four sides are equal

Side: Any side of the square (all are equal)

Given
Side = 5 units
Perimeter
P = 4s
Area
A = s²
Results
P = 20, A = 25
Step 1: Identify the dimensions

Since it's a square, all sides = 5 units

Length = 5 units, Width = 5 units

Step 2: Calculate the perimeter

Method 1: P = 2(l + w) = 2(5 + 5) = 2(10) = 20

Method 2: P = 4s = 4(5) = 20 (since all sides are equal)

Step 3: Calculate the area

Method 1: A = l × w = 5 × 5 = 25

Method 2: A = s² = 5² = 25 (since all sides are equal)

Step 4: State both answers

Perimeter = 20 units

Area = 25 square units

Perimeter = 20 units, Area = 25 square units
Final answer:

The perimeter is 20 units and the area is 25 square units

Applied rules:

• Square Property: All four sides are equal

• Square Perimeter: P = 4s where s is the side length

• Square Area: A = s²

• Rectangle formulas: Also work for squares

3 Different dimensions
Exercise 3
Find the perimeter and area of a rectangle with length 8 units and width 3 units.
Definition:

Longer vs Shorter sides: Length is typically the longer side, width the shorter

Given
Length = 8, Width = 3
Perimeter
P = 2(l+w)
Area
A = l×w
Results
P = 22, A = 24
Step 1: Identify the dimensions

Length = 8 units, Width = 3 units

Step 2: Calculate the perimeter

Perimeter = 2 × (length + width)

Perimeter = 2 × (8 + 3)

Perimeter = 2 × 11 = 22 units

Step 3: Calculate the area

Area = length × width

Area = 8 × 3 = 24 square units

Step 4: Verify with addition method

Perimeter = 8 + 3 + 8 + 3 = 22 units ✓

Perimeter = 22 units, Area = 24 square units
Final answer:

The perimeter is 22 units and the area is 24 square units

Applied rules:

• Rectangle Property: Opposite sides are equal

• Perimeter Formula: P = 2(l+w)

• Area Formula: A = l×w

• Verification: Adding all four sides gives the same perimeter

Rules and methods, laws,...
\(P = 2(l + w), A = l \times w\)
Perimeter and Area of Rectangle
Rectangle
\(4 \text{ sides}, \text{opposite equal}\)
Properties
Perimeter
\(P = 2(l + w)\)
Rectangle formula
Area
\(A = l \times w\)
Rectangle formula
Square
\(P = 4s, A = s^2\)
All sides equal
Key definitions:

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

Perimeter: The total distance around the outside of a shape

Area: The amount of space inside a two-dimensional shape

Length: Usually the longer side of a rectangle

Width: Usually the shorter side of a rectangle

Side: Any edge of the rectangle

Complete methodology:
  1. Identify the shape: Confirm it's a rectangle or square
  2. Measure dimensions: Find the length and width
  3. Select formulas: Use P = 2(l + w) for perimeter, A = l × w for area
  4. Substitute values: Put the length and width into the formulas
  5. Calculate perimeter: Add length and width, then multiply by 2
  6. Calculate area: Multiply length by width
  7. Verify: Check by adding all four sides for perimeter
Tip 1: Remember that opposite sides of a rectangle are equal.
Tip 2: A square is a special rectangle where all sides are equal.
Tip 3: You can always check perimeter by adding all four sides together.
Tip 4: Don't forget to include the unit of measurement in your answer.
Common errors: Forgetting to multiply by 2 for perimeter, confusing perimeter with area, misidentifying length vs width.
Memory aids: "Perimeter is the fence around the yard" and "Area is the space inside".
Formulas to know by heart:

• Perimeter of rectangle: P = 2(l + w)

• Area of rectangle: A = l × w

• Perimeter of square: P = 4s

• Area of square: A = s²

Solution: Exercises 4 to 5
4 Real-world application
Exercise 4
A rectangular room has a length of 10 feet and a width of 8 feet. How much carpet is needed to cover the floor? How much baseboard is needed to go around the walls?
Definition:

Carpet: Material needed to cover the floor, which equals the area

Baseboard: Trim that goes around the walls at floor level, which equals the perimeter

Given
Length = 10 ft, Width = 8 ft
Area
A = l×w
Perimeter
P = 2(l+w)
Results
A = 80 ft², P = 36 ft
Step 1: Identify what's being asked

Part 1: How much carpet is needed? This requires area (covering the floor)

Part 2: How much baseboard is needed? This requires perimeter (going around the walls)

Step 2: Identify the dimensions

Room length = 10 feet, Room width = 8 feet

Step 3: Calculate the area (for carpet)

Area = length × width = 10 × 8 = 80 square feet

So, 80 square feet of carpet is needed

Step 4: Calculate the perimeter (for baseboard)

Perimeter = 2 × (length + width) = 2 × (10 + 8) = 2 × 18 = 36 feet

So, 36 feet of baseboard is needed

Step 5: State both answers in context

80 square feet of carpet is needed for the floor

36 feet of baseboard is needed for the walls

Carpet needed = 80 sq ft, Baseboard needed = 36 ft
Final answer:

80 square feet of carpet is needed to cover the floor, and 36 feet of baseboard is needed to go around the walls

Applied rules:

• Real-world modeling: Carpet covers area, baseboard follows perimeter

• Area application: Covering surfaces requires area measurement

• Perimeter application: Borders/trims require perimeter measurement

5 Comparison problem
Exercise 5
Rectangle A has length 7 cm and width 5 cm. Rectangle B has length 6 cm and width 6 cm. Which rectangle has a larger area? Which has a larger perimeter?
Definition:

Comparison problem: Calculating and comparing measurements of different shapes

Rectangle A: Standard rectangle with different length and width

Rectangle B: Actually a square since length = width

Rectangle A
P = 2(7+5)=24cm, A = 7×5=35cm²
Rectangle B
P = 2(6+6)=24cm, A = 6×6=36cm²
Results
Same P, B has larger A
Step 1: Calculate Rectangle A's perimeter and area

Rectangle A: length = 7 cm, width = 5 cm

Perimeter = 2 × (7 + 5) = 2 × 12 = 24 cm

Area = 7 × 5 = 35 square cm

Step 2: Calculate Rectangle B's perimeter and area

Rectangle B: length = 6 cm, width = 6 cm (this is actually a square)

Perimeter = 2 × (6 + 6) = 2 × 12 = 24 cm

Area = 6 × 6 = 36 square cm

Step 3: Compare the perimeters

Rectangle A: 24 cm

Rectangle B: 24 cm

24 cm = 24 cm, so they have the same perimeter

Step 4: Compare the areas

Rectangle A: 35 square cm

Rectangle B: 36 square cm

36 square cm > 35 square cm, so Rectangle B has the larger area

Step 5: State the conclusions

Both rectangles have the same perimeter (24 cm)

Rectangle B has the larger area (36 square cm vs 35 square cm)

Same perimeter (24 cm), Rectangle B has larger area (36 cm² vs 35 cm²)
Final answer:

Both rectangles have the same perimeter of 24 cm, but Rectangle B has a larger area of 36 square cm compared to Rectangle A's area of 35 square cm

Applied rules:

• Comparison strategy: Calculate both values before comparing

• Shape identification: Recognizing that Rectangle B is a square

• Units consistency: Both measurements in the same units

Complete Guide: Laws, Methods, Rules, Definitions, Expressions
\(P = 2(l + w), A = l \times w\)
Perimeter and Area of Rectangle
Key definitions:

Rectangle: A four-sided shape (quadrilateral) with opposite sides equal in length and all interior angles measuring 90° (right angles).

Perimeter: The total distance around the outside edge of a two-dimensional shape, measured in linear units.

Area: The amount of space inside a two-dimensional shape, measured in square units.

Length: Typically the longer side of a rectangle, though this can vary depending on orientation.

Width: Typically the shorter side of a rectangle, though this can vary depending on orientation.

Side: Any of the four line segments that form the boundary of the rectangle.

Complete methodology:
  1. Identify the shape: Confirm that the figure is indeed a rectangle
  2. Measure or identify dimensions: Determine the length and width of the rectangle
  3. Select the appropriate formulas: Use P = 2(l + w) for perimeter, A = l × w for area
  4. Substitute values: Replace l (length) and w (width) with their actual measurements
  5. Perform perimeter calculations: Add length and width, then multiply by 2
  6. Perform area calculations: Multiply length by width
  7. State the answers: Include the appropriate units of measurement
  8. Verify the results: Optionally add all four sides to confirm the perimeter
Tip 1: Remember that rectangles have two pairs of equal sides. If you know the length and width, you automatically know all four sides.
Tip 2: A square is a special type of rectangle where all four sides are equal. You can use either P = 2(l + w) or P = 4s for squares.
Tip 3: Always include the unit of measurement (inches, feet, centimeters, etc.) in your final answer. Use square units for area (in², ft², cm²).

Tip 4: You can check your perimeter answer by adding all four sides individually: length + width + length + width.
Rectangle Properties: 4 right angles, opposite sides equal, diagonals equal and bisect each other, sum of interior angles is 360°.
Perimeter Characteristics: Measures the boundary of a shape, always a linear measurement, expressed in units like inches, feet, or centimeters.
Area Characteristics: Measures the space inside a shape, always a square measurement, expressed in square units like square inches, square feet, or square centimeters.
Common Errors: Confusing perimeter with area, forgetting to multiply by 2 in perimeter formula, misidentifying length vs width, forgetting square units for area.
Essential formulas:

• Perimeter of rectangle: P = 2(l + w)

• Area of rectangle: A = l × w

• Perimeter of square: P = 4s

• Area of square: A = s²

• Diagonal of rectangle: d = √(l² + w²)

Perimeter and Area of Rectangles Infographic
📏
Rectangle Structure

• 4 sides: 2 lengths (l) + 2 widths (w)

• 4 right angles (90°)

• Opposite sides equal

• Perimeter = 2(l + w)

• Area = l × w

Perimeter Calculation

• Method 1: Add all sides

• Method 2: P = 2(l + w)

• For squares: P = 4s

• Units: same as side measurements

Area Calculation

• A = l × w

• For squares: A = s²

• Units: square measurements

• Measures space inside

Verification

• Add all 4 sides: l + w + l + w

• Should equal 2(l + w)

• Check arithmetic carefully

• Include units in answer

Problem-Solving Workflow
1
Identify Shape
2
Get Dimensions
3
Apply Formulas
4
Calculate
5
Verify Answer
Visual Representation

Rectangle with length (l) and width (w)

Perimeter = distance around the rectangle

Area = space inside the rectangle

P = 2(l + w), A = l × w

Remember: Perimeter measures the boundary, Area measures the space inside!

Questions & Answers

Question: What's the difference between perimeter and area? I always get them mixed up.

Answer: This is a very common question! Here's the key difference:

  • Perimeter: The distance around the outside of a shape. Think of it as walking around the edge of a garden. It's measured in linear units (inches, feet, centimeters).
  • Area: The amount of space inside a shape. Think of it as covering the floor with tiles. It's measured in square units (square inches, square feet, square centimeters).

For a rectangle: Perimeter = 2(length + width) [linear measurement] Area = length × width [square measurement]

A helpful memory trick: "Perimeter" starts with "P" like "Periscope" - you're looking around the edge. "Area" sounds like "are" - "How much area is covered?"

Question: Why do we multiply by 2 in the perimeter formula? Can't we just add the length and width once?

Answer: Great question! We multiply by 2 because a rectangle has TWO lengths and TWO widths. Let's break it down:

  • A rectangle has 4 sides: top, bottom, left, right
  • The top and bottom sides are both equal to the length
  • The left and right sides are both equal to the width
  • So the total perimeter is: length + length + width + width = 2×length + 2×width = 2(length + width)

If you only added length + width, you'd only be counting half the perimeter! For example, if length = 4 and width = 3: Using P = l + w: 4 + 3 = 7 (incorrect!) Using P = 2(l + w): 2(4 + 3) = 2(7) = 14 (correct!)

You can verify: 4 + 3 + 4 + 3 = 14 ✓

Question: Is a square really a rectangle? I thought they were different shapes.

Answer: Yes, a square IS a special type of rectangle! Here's why:

  • A rectangle is defined as a four-sided shape with all angles equal to 90° (right angles)
  • A square meets this definition: it has four sides and all four angles are 90°
  • But a square has an extra property: all four sides are equal in length

Think of it this way: all squares are rectangles, but not all rectangles are squares. It's like how all dogs are animals, but not all animals are dogs.

This is why the rectangle formulas P = 2(l + w) and A = l × w work for squares too. If all sides are equal (let's call that length 's'), then P = 2(s + s) = 2(2s) = 4s, and A = s × s = s²!