Using Formulas to Calculate Area and Perimeter in Grade 3

Master area and perimeter formulas: rectangles, squares, calculations with detailed explanations through these 5 exercises.

Solution: Exercises 1 to 3
1 Rectangle area and perimeter
Exercise 1
Find the area and perimeter of a rectangle with length 8 cm and width 5 cm.
Definition:

Rectangle: A shape with 4 sides and 4 right angles

Length: The longer side of the rectangle

Width: The shorter side of the rectangle

Area: The space inside the rectangle

Perimeter: The distance around the rectangle

Calculation method:
  1. Identify the length and width of the rectangle
  2. Use the area formula: Area = length × width
  3. Use the perimeter formula: Perimeter = 2 × (length + width)
  4. Calculate both measurements
  5. State the answers with correct units
Area Formula
A = l × w
Perimeter Formula
P = 2(l + w)
Results
A=40, P=26
Step 1: Identify the measurements

Length = 8 cm, Width = 5 cm

Step 2: Calculate the area

Area = length × width = 8 × 5 = 40 cm²

Step 3: Calculate the perimeter

Perimeter = 2 × (length + width) = 2 × (8 + 5) = 2 × 13 = 26 cm

Step 4: State the final answers

Area = 40 cm², Perimeter = 26 cm

Area = 40 cm², Perimeter = 26 cm
Final answer:

The area of the rectangle is 40 square centimeters and the perimeter is 26 centimeters.

Applied rules:

• Rectangle Area Formula: Area = length × width

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

• Multiplication: Calculate length × width for area

• Addition: Add length + width first, then multiply by 2 for perimeter

• Units: Area in square units (cm²), Perimeter in linear units (cm)

Tip 1: Area measures space inside, perimeter measures the boundary.
Tip 2: Perimeter formula can also be written as P = l + l + w + w.
2 Square area and perimeter
Exercise 2
Find the area and perimeter of a square with side length 6 inches.
Definition:

Square: A rectangle with all sides equal

Side: Each edge of the square

Equal sides: All four sides have the same length

Square Area
A = s²
Square Perimeter
P = 4s
Results
A=36, P=24
Step 1: Identify the side length

Side = 6 inches

Step 2: Calculate the area

Area = side × side = 6 × 6 = 36 in²

Step 3: Calculate the perimeter

Perimeter = 4 × side = 4 × 6 = 24 inches

Step 4: State the final answers

Area = 36 in², Perimeter = 24 inches

Area = 36 in², Perimeter = 24 inches
Final answer:

The area of the square is 36 square inches and the perimeter is 24 inches.

Applied rules:

• Square Area Formula: Area = side × side = s²

• Square Perimeter Formula: Perimeter = 4 × side = 4s

• Special case: Square is a rectangle with equal sides

• Squaring: Side² means side × side

• Multiplication: Perimeter = 4 × one side length

Tip 1: A square is a special type of rectangle.
Tip 2: For squares, just remember: Area = side², Perimeter = 4 × side.
3 Finding missing dimension
Exercise 3
A rectangle has an area of 24 cm² and a width of 4 cm. What is its length? What is its perimeter?
Definition:

Missing dimension: When one measurement is unknown in a formula

Algebra: Using formulas backwards to find unknown values

Area Formula
A = l × w
Known Values
A = 24, w = 4
Find
l = ?, P = ?
Step 1: Write the area formula

Area = length × width

Step 2: Substitute known values

24 = length × 4

Step 3: Solve for length

length = 24 ÷ 4 = 6 cm

Step 4: Calculate the perimeter

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

Step 5: Verify the answer

Area = 6 × 4 = 24 cm² ✓

Length = 6 cm, Perimeter = 20 cm
Final answer:

The length of the rectangle is 6 cm and the perimeter is 20 cm.

Applied rules:

• Algebra: Rearrange formulas to solve for unknown values

• Division: To find length: Area ÷ width

• Perimeter formula: Once you have length, calculate perimeter

• Verification: Check your answer by substituting back

Tip 1: When finding missing dimensions, rearrange the formula.
Tip 2: Always verify your answer by checking it in the original formula.
Area and Perimeter Formulas and Methods
Rectangle: A = l × w, P = 2(l + w)
Rectangle Formulas
Square: A = s², P = 4s
Square Formulas
Rectangle
4 sides, 4 right angles
Opposite sides equal
Square
4 equal sides
All sides the same
Area
Space inside
Measured in squares
Perimeter
Distance around
Boundary length
Key definitions:

Rectangle: A quadrilateral with 4 right angles and opposite sides equal

Square: A rectangle with all 4 sides equal

Area: The amount of space inside a 2D shape

Perimeter: The total distance around the outside of a shape

Length: The longer side of a rectangle

Width: The shorter side of a rectangle

Side: Each edge of a square

Complete methodology:
  1. Identify the shape: Determine if it's a rectangle or square
  2. Find measurements: Locate length, width, or side lengths
  3. Choose formula: Select area or perimeter formula
  4. Substitute values: Replace variables with numbers
  5. Calculate: Perform the arithmetic operations
  6. Check units: Area in square units, perimeter in linear units
Tip 1: Area measures space inside, perimeter measures the boundary.
Tip 2: For squares, all sides are equal, so Area = s² and P = 4s.
Tip 3: Always include units in your final answer.
Tip 4: If one dimension is missing, rearrange the formula to solve.
Common errors: Confusing area and perimeter, forgetting to square units for area, wrong formulas.
Memory aid: "Area = space inside (multiply), Perimeter = walk around (add)".
Formulas to know by heart:

• Rectangle Area: A = length × width

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

• Square Area: A = side × side = s²

• Square Perimeter: P = 4 × side = 4s

Solution: Exercises 4 to 5
4 Real-world application
Exercise 4
Tom wants to put a fence around his rectangular garden that is 10 feet long and 7 feet wide. He also wants to cover the garden with mulch. How much fencing does he need? How much mulch does he need?
Definition:

Real-world problem: Applying math to practical situations

Fencing: Goes around the perimeter of the garden

Mulch: Covers the area of the garden

Given
l = 10 ft, w = 7 ft
Fencing (P)
P = 2(l + w)
Mulch (A)
A = l × w
Step 1: Identify what's needed

Fencing = perimeter, Mulch = area

Step 2: Calculate the perimeter

Perimeter = 2 × (length + width) = 2 × (10 + 7) = 2 × 17 = 34 feet

Step 3: Calculate the area

Area = length × width = 10 × 7 = 70 square feet

Step 4: State the answer

Tom needs 34 feet of fencing and 70 square feet of mulch

Fencing: 34 ft, Mulch: 70 ft²
Final answer:

Tom needs 34 feet of fencing and 70 square feet of mulch.

Applied rules:

• Problem solving: Identify what quantity is needed (area vs perimeter)

• Perimeter formula: P = 2 × (length + width)

• Area formula: A = length × width

• Real-world context: Fencing goes around (perimeter), mulch covers inside (area)

Tip 1: Read word problems carefully to determine area or perimeter.
Tip 2: Words like "around," "fence," "border" suggest perimeter.
5 Comparing shapes
Exercise 5
Rectangle A has length 8 cm and width 5 cm. Rectangle B has length 6 cm and width 7 cm. Which rectangle has the larger area? Which has the larger perimeter?
Definition:

Comparison: Finding which value is greater or lesser

Multiple calculations: Solving more than one problem

Rectangle A
A=40, P=26
Rectangle B
A=42, P=26
Comparison
B>A, P equal
Step 1: Calculate Rectangle A's area and perimeter

Area = 8 × 5 = 40 cm², Perimeter = 2 × (8 + 5) = 26 cm

Step 2: Calculate Rectangle B's area and perimeter

Area = 6 × 7 = 42 cm², Perimeter = 2 × (6 + 7) = 26 cm

Step 3: Compare the areas

42 cm² > 40 cm², so Rectangle B has the larger area

Step 4: Compare the perimeters

Both perimeters are 26 cm, so they are equal

Step 5: State the conclusions

Rectangle B has the larger area, both have equal perimeters

Area: B > A, Perimeter: A = B
Final answer:

Rectangle B has the larger area (42 cm² vs 40 cm²), but both rectangles have equal perimeters (26 cm each).

Applied rules:

• Comparison: Calculate each measurement separately, then compare

• Multiple calculations: Solve each rectangle individually

• Area formula: Area = length × width

• Perimeter formula: Perimeter = 2 × (length + width)

Tip 1: Organize your work when comparing multiple shapes.
Tip 2: Different rectangles can have the same perimeter but different areas.
Area and Perimeter Laws, Methods, Rules, and Definitions
Rectangle: A = l × w, P = 2(l + w)
Rectangle Formulas
Square: A = s², P = 4s
Square Formulas
Key definitions:

Rectangle: A quadrilateral with 4 right angles and opposite sides equal

Square: A rectangle with all 4 sides equal

Area: The amount of space inside a 2D shape, measured in square units

Perimeter: The total distance around the outside of a shape, measured in linear units

Length: The longer side of a rectangle

Width: The shorter side of a rectangle

Side: Each edge of a square

Square units: Units like cm², in², ft² that represent area

Linear units: Units like cm, in, ft that represent length/perimeter

Complete methodology:
  1. Identify the shape: Determine if it's a rectangle or square
  2. Find the measurements: Locate length, width, or side lengths
  3. Choose the correct formula: Select area or perimeter formula
  4. Substitute values: Replace variables with actual measurements
  5. Perform calculations: Follow order of operations (parentheses first)
  6. State the answer: Include correct units (square for area, linear for perimeter)
Tip 1: Area measures space inside (think "carpet"), perimeter measures the boundary (think "fence").
Tip 2: For perimeter of rectangles, you can add all 4 sides: l + w + l + w.
Tip 3: Squares are special rectangles where length = width.
Tip 4: Always check that your units match (cm² for area, cm for perimeter).
Common errors: Forgetting to square units for area, confusing area and perimeter, misapplying formulas.
Memory aid: "Area = multiply (inside), Perimeter = add (around)".
Formulas to know by heart:

• Rectangle Area: A = length × width

• Rectangle Perimeter: P = 2 × (length + width) or P = l + w + l + w

• Square Area: A = side × side = s²

• Square Perimeter: P = 4 × side = 4s

• Unit conversion: Area units are squared (cm², in², ft²), Perimeter units are linear (cm, in, ft)

Rectangle
4 sides, 4 right angles
Opposite sides equal
Square
4 equal sides
All sides the same
Area
Space inside
Measured in squares
Perimeter
Distance around
Boundary length

Detailed Summary: Area and Perimeter Formulas

Key Definitions

Rectangle

A rectangle is a four-sided shape with four right angles (90°) and opposite sides that are equal in length. It has two pairs of parallel sides - the top and bottom sides are equal, and the left and right sides are equal.

Square

A square is a special type of rectangle where all four sides are equal in length. It has four right angles and all sides are the same length. Every square is also a rectangle, but not every rectangle is a square.

Area

Area is the amount of space inside a two-dimensional shape. It measures how many square units fit inside the shape. Area is always measured in square units (cm², in², ft², etc.).

Perimeter

Perimeter is the total distance around the outside of a shape. It measures the length of the boundary of the shape. Perimeter is always measured in linear units (cm, in, ft, etc.).

Length and Width

Length is typically the longer side of a rectangle, while width is the shorter side. These are the two measurements needed to calculate area and perimeter of a rectangle.

Core Rules and Principles

Rectangle Area Rule

Area of a rectangle = length × width

This rule works because you can think of a rectangle as being made up of rows and columns of unit squares. Multiplying length by width tells you how many unit squares fit inside.

Rectangle Perimeter Rule

Perimeter of a rectangle = 2 × (length + width) or length + width + length + width

This rule adds up all four sides of the rectangle. Since opposite sides are equal, you can add length + width and multiply by 2.

Square Area Rule

Area of a square = side × side = side²

Since all sides of a square are equal, you only need to know one side length to calculate the area.

Square Perimeter Rule

Perimeter of a square = 4 × side

Since all four sides of a square are equal, you can multiply one side by 4 to get the total perimeter.

Step-by-Step Methods

Method for Calculating Area and Perimeter

  1. Identify the shape: Determine if it's a rectangle or square
  2. Measure or find the dimensions: Get the length and width (for rectangle) or side length (for square)
  3. Select the appropriate formula: Choose area or perimeter formula based on what you need
  4. Substitute values: Replace the variables in the formula with the actual measurements
  5. Calculate: Perform the arithmetic operations (remember order of operations)
  6. Check units: Make sure your answer has the correct units (square for area, linear for perimeter)
  7. Verify: Check your answer makes sense in the context of the problem

Examples

Simple Example

Rectangle with length = 5 cm and width = 3 cm:

Area = 5 × 3 = 15 cm²

Perimeter = 2 × (5 + 3) = 2 × 8 = 16 cm

Advanced Example

Square with side length = 7 inches:

Area = 7² = 49 in²

Perimeter = 4 × 7 = 28 inches

Word Problem Example

Garden plot is 12 feet long and 8 feet wide. Fence costs $3 per foot. How much will fencing cost?

Perimeter = 2 × (12 + 8) = 40 feet

Cost = 40 × $3 = $120

Tips and Tricks

Tips for Success

  • Remember the difference: Area = space inside (multiply), Perimeter = distance around (add)
  • Draw a picture: Visualizing the shape helps you understand what you're measuring
  • Check your units: Area should always be in square units, perimeter in linear units
  • Double-check calculations: Mistakes in multiplication or addition are common
  • Estimate first: Make sure your answer makes sense (a small rectangle shouldn't have a huge area)

Common Pitfalls to Avoid

  • Confusing area and perimeter: Reading problems carefully to determine which measurement is needed
  • Forgetting to square units: Area answers must have square units (cm², not cm)
  • Order of operations: Calculate inside parentheses first in perimeter formula
  • Mixing up length and width: Pay attention to which measurement is which

Key Notes for Memorization

Memory Aids

  • "Area" starts with "A" like "All" - you count all the squares inside
  • "Perimeter" starts with "P" like "Path" - you walk the path around the outside
  • Squares are special: Area = s², Perimeter = 4s (just need one measurement)
  • Rectangles need two: Area = l × w, Perimeter = 2(l + w)

Quick Reference

Shape Area Formula Perimeter Formula
Rectangle A = l × w P = 2(l + w)
Square A = s² P = 4s

Questions & Answers

Question: Why do we say area is in square units and perimeter is in regular units? What's the difference?

Answer: Great question! The difference comes from what we're measuring:

Area measures how much space is inside a shape. Think of covering the shape with tiny squares - you're counting how many little squares fit inside. Since you're measuring in two directions (length and width), you use square units: square centimeters (cm²), square inches (in²), etc.

Perimeter measures the distance around the outside of a shape. It's like walking along the edge of the shape - you're measuring just one direction (the boundary). So you use regular units: centimeters (cm), inches (in), feet (ft), etc.

For example, if you have a rectangle that's 4 cm by 3 cm:

  • Area = 4 × 3 = 12 cm² (covers 12 squares inside)
  • Perimeter = 2(4 + 3) = 14 cm (distance around the edge)

Question: Can two different rectangles have the same area but different perimeters? Or the same perimeter but different areas?

Answer: Yes, absolutely! This is a fascinating property of rectangles:

Same area, different perimeters:

  • Rectangle A: 6 cm × 4 cm → Area = 24 cm², Perimeter = 20 cm
  • Rectangle B: 8 cm × 3 cm → Area = 24 cm², Perimeter = 22 cm
Both have the same area (24 cm²) but different perimeters!

Same perimeter, different areas:

  • Rectangle C: 7 cm × 3 cm → Area = 21 cm², Perimeter = 20 cm
  • Rectangle D: 6 cm × 4 cm → Area = 24 cm², Perimeter = 20 cm
Both have the same perimeter (20 cm) but different areas!

This shows that area and perimeter are independent measurements - knowing one doesn't tell you the other!

Question: How can I remember which formula is for area and which is for perimeter? They both use length and width.

Answer: Here are some memory tricks to help you remember:

For Area:

  • Think "A" for "All" - you multiply to count all the squares inside
  • Area = Multiply (A-M)
  • Think of carpeting a room - you multiply length × width

For Perimeter:

  • Think "P" for "Path" - you add to walk around the path
  • Perimeter = Add (P-A)
  • Think of fencing a yard - you add up all the sides

Also remember: Area uses multiplication and results in square units (cm²), while Perimeter uses addition and results in linear units (cm).