Solved Exercises on Finding Perimeter of Rectangle with Area in Grade 3

Master finding perimeter of rectangles when given area: basic formulas, geometric relationships, real-life applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic rectangle
Exercise 1
A rectangle has an area of 12 square units and one side of 3 units. Find the perimeter.
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

Method when given area:
  1. Use the area formula: Area = length × width
  2. Find the missing dimension: width = Area ÷ length
  3. Calculate perimeter: Perimeter = 2 × (length + width)
Given
Area = 12, Side = 3
Find Other Side
12 ÷ 3 = 4
Perimeter
P = 2(3+4) = 14
Step 1: Identify what's given

Area = 12 square units, one side = 3 units

Step 2: Find the missing dimension

Using Area = length × width

If one side is 3 units, then: 12 = 3 × other side

Other side = 12 ÷ 3 = 4 units

Step 3: Calculate the perimeter

Perimeter = 2 × (length + width)

Perimeter = 2 × (3 + 4) = 2 × 7 = 14 units

Step 4: Verify the solution

Area = 3 × 4 = 12 square units ✓

Perimeter = 3 + 4 + 3 + 4 = 14 units ✓

Perimeter = 14 units
Final answer:

The perimeter of the rectangle is 14 units

Applied rules:

• Area formula: A = l × w

• Division to find missing dimension: w = A ÷ l

• Perimeter formula: P = 2(l + w)

2 Square example
Exercise 2
A square has an area of 25 square units. Find the perimeter.
Definition:

Square: A special rectangle where all four sides are equal

Area of square: A = s² where s is the side length

Given
Area = 25
Find Side
√25 = 5
Perimeter
P = 4×5 = 20
Step 1: Identify what's given

It's a square with area = 25 square units

Step 2: Find the side length

For a square: Area = side²

25 = side²

side = √25 = 5 units

Step 3: Calculate the perimeter

Method 1: P = 4 × side = 4 × 5 = 20

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

Step 4: Verify the solution

Area = 5 × 5 = 25 square units ✓

Perimeter = 5 + 5 + 5 + 5 = 20 units ✓

Perimeter = 20 units
Final answer:

The perimeter of the square is 20 units

Applied rules:

• Square area: A = s²

• Square side: s = √A

• Square perimeter: P = 4s

3 Different dimensions
Exercise 3
A rectangle has an area of 18 square units and one side of 6 units. Find the perimeter.
Definition:

Rectangle relationship: Area = length × width

Perimeter: P = 2(length + width)

Given
Area = 18, Side = 6
Find Other Side
18 ÷ 6 = 3
Perimeter
P = 2(6+3) = 18
Step 1: Identify what's given

Area = 18 square units, one side = 6 units

Step 2: Find the missing dimension

Using Area = length × width

18 = 6 × other side

Other side = 18 ÷ 6 = 3 units

Step 3: Calculate the perimeter

Perimeter = 2 × (length + width)

Perimeter = 2 × (6 + 3) = 2 × 9 = 18 units

Step 4: Verify the solution

Area = 6 × 3 = 18 square units ✓

Perimeter = 6 + 3 + 6 + 3 = 18 units ✓

Perimeter = 18 units
Final answer:

The perimeter of the rectangle is 18 units

Applied rules:

• Area formula: A = l × w

• Division to find missing dimension: w = A ÷ l

• Perimeter formula: P = 2(l + w)

Rules and methods, laws,...
\(A = l \times w, P = 2(l + w)\)
Area and Perimeter of Rectangle
Rectangle
\(A = l \times w\)
Area formula
Perimeter
\(P = 2(l + w)\)
Perimeter formula
Square
\(A = s^2, P = 4s\)
Special case
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. Identify what's given: Area and one dimension
  3. Find missing dimension: Use Area ÷ known dimension
  4. Calculate perimeter: Use P = 2(l + w)
  5. Verify: Check that Area = l × w
Tip 1: When you know area and one side, divide area by that side to get the other side.
Tip 2: A square is a special rectangle where all sides are equal.
Tip 3: You can always check by multiplying length × width to verify the area.
Tip 4: Don't forget to include the unit of measurement in your answer.
Common errors: Forgetting to find the missing dimension, confusing perimeter with area, misidentifying length vs width.
Memory aids: "Area is length times width" and "Perimeter is the fence around the yard".
Formulas to know by heart:

• Area of rectangle: A = l × w

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

• Area of square: A = s²

• Perimeter of square: P = 4s

• Finding missing side: w = A ÷ l

Solution: Exercises 4 to 5
4 Real-world application
Exercise 4
A rectangular room has an area of 30 square feet and a width of 5 feet. How much baseboard is needed to go around the walls?
Definition:

Word problem: A math problem presented in a real-world context

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

Given
Area = 30 ft², Width = 5 ft
Find Length
30 ÷ 5 = 6 ft
Perimeter
P = 2(6+5) = 22 ft
Step 1: Identify what's being asked

The problem asks for the amount of baseboard needed to go around the walls

This means we need to find the perimeter of the rectangular room

Step 2: Identify what's given

Area = 30 square feet, Width = 5 feet

Step 3: Find the missing dimension

Using Area = length × width

30 = length × 5

Length = 30 ÷ 5 = 6 feet

Step 4: Calculate the perimeter

Perimeter = 2 × (length + width)

Perimeter = 2 × (6 + 5) = 2 × 11 = 22 feet

Step 5: State the answer in context

22 feet of baseboard is needed to go around the walls

Baseboard needed = 22 feet
Final answer:

22 feet of baseboard is needed to go around the walls

Applied rules:

• Real-world modeling: Representing physical situations with geometric concepts

• Area to find missing dimension: Using division to find the unknown side

• Perimeter application: Perimeter represents the distance around a shape

5 Comparison problem
Exercise 5
Rectangle A has area 24 square cm and length 6 cm. Rectangle B has area 20 square cm and length 5 cm. Which rectangle has a larger perimeter?
Definition:

Comparison problem: Calculating and comparing measurements of different shapes

Rectangle A: Area and length given, find width

Rectangle B: Area and length given, find width

Rectangle A
W = 24÷6=4, P = 2(6+4)=20
Rectangle B
W = 20÷5=4, P = 2(5+4)=18
Result
A larger by 2
Step 1: Calculate Rectangle A's dimensions and perimeter

Rectangle A: Area = 24 cm², Length = 6 cm

Width = Area ÷ Length = 24 ÷ 6 = 4 cm

Perimeter = 2 × (length + width) = 2 × (6 + 4) = 2 × 10 = 20 cm

Step 2: Calculate Rectangle B's dimensions and perimeter

Rectangle B: Area = 20 cm², Length = 5 cm

Width = Area ÷ Length = 20 ÷ 5 = 4 cm

Perimeter = 2 × (length + width) = 2 × (5 + 4) = 2 × 9 = 18 cm

Step 3: Compare the perimeters

Rectangle A: 20 cm

Rectangle B: 18 cm

20 cm > 18 cm

Step 4: State the conclusion

Rectangle A has the larger perimeter of 20 cm compared to Rectangle B's 18 cm

Rectangle A has larger perimeter (20 cm vs 18 cm)
Final answer:

Rectangle A has a larger perimeter of 20 cm compared to Rectangle B's perimeter of 18 cm

Applied rules:

• Comparison strategy: Calculate both values before comparing

• Area to find missing dimension: Using division to find the unknown width

• Units consistency: Both measurements in the same unit

Complete Guide: Laws, Methods, Rules, Definitions, Expressions
\(A = l \times w, P = 2(l + w)\)
Area and Perimeter 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. Identify given information: Determine what is known (area and one dimension)
  3. Find missing dimension: Use A = l × w to find the unknown side
  4. Calculate perimeter: Use P = 2(l + w) with both dimensions
  5. State the answer: Include the appropriate unit of measurement
  6. Verify the result: Check that the area equals length × width
Tip 1: Remember that rectangles have two pairs of equal sides. If you know the area and one side, you can find the other side.
Tip 2: A square is a special type of rectangle where all four sides are equal. You can use either A = l × w or A = s² for squares.
Tip 3: Always include the unit of measurement (inches, feet, centimeters, etc.) in your final answer.
Tip 4: You can check your answer by multiplying length by width to verify the area.
Rectangle Properties: 4 right angles, opposite sides equal, diagonals equal and bisect each other, sum of interior angles is 360°.
Area Characteristics: Measures the space inside a shape, always a square measurement, expressed in square units.
Common Errors: Confusing perimeter with area, forgetting to multiply by 2 in perimeter formula, misidentifying length vs width.
Essential formulas:

• Area of rectangle: A = l × w

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

• Area of square: A = s²

• Perimeter of square: P = 4s

• Finding missing dimension: w = A ÷ l or l = A ÷ w

Rectangle Area-Perimeter Relationships
📐
Rectangle Structure

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

• 4 right angles (90°)

• Opposite sides equal

• Area = l × w

• Perimeter = 2(l + w)

Area-Perimeter Connection

• Given area and one side

• Find missing side: w = A ÷ l

• Then calculate perimeter

• P = 2(l + w)

Verification

• Check: Area = l × w

• Check: Perimeter = sum of all sides

• Include units in answer

• Verify calculations

Problem-Solving Workflow
1
Identify Given
2
Find Missing Side
3
Calculate Perimeter
4
Verify Answer
Visual Representation

Rectangle with length (l) and width (w)

Area = l × w

Perimeter = 2(l + w)

Relationship: A = l × w → find missing dimension

Remember: Area and perimeter are related through dimensions!

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 divide the area by one side to find the other side?

Answer: This comes from the area formula! For a rectangle:

  • Area = length × width
  • If you know the area and one side, you can rearrange this equation
  • Area = length × width → width = Area ÷ length
  • Or: Area = length × width → length = Area ÷ width

For example, if area = 12 and one side = 3: 12 = 3 × other side 12 ÷ 3 = other side 4 = other side

This is the inverse operation - if multiplication gives us area, then division helps us find a missing dimension.

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