Review and Application of Area, Perimeter, and Shapes: Comprehensive Practice

Master area, perimeter, and shapes through comprehensive review and application exercises with detailed solutions and visual aids.

Solution: Exercises 1 to 3
1 Rectangle Review
Exercise 1
Find the area and perimeter of a rectangle with length 8 cm and width 5 cm. Then draw a square with the same area. What is the perimeter of the square?
Definition:

Area: The space inside a shape. Perimeter: The distance around the outside of a shape.

Calculation Method:
  1. Calculate Rectangle Area: length × width
  2. Calculate Rectangle Perimeter: 2 × (length + width)
  3. Find Square Side: √area of rectangle
  4. Calculate Square Perimeter: 4 × side length
Rectangle
8×5=40 cm²
Perimeter
2(8+5)=26 cm
Square side
√40≈6.32 cm
Step 1: Calculate rectangle area

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

Step 2: Calculate rectangle perimeter

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

Step 3: Find square side length

For same area (40 cm²), square side = √40 ≈ 6.32 cm

Step 4: Calculate square perimeter

Perimeter = 4 × side = 4 × 6.32 ≈ 25.28 cm

Rectangle: Area = 40 cm², Perimeter = 26 cm
Square: Side ≈ 6.32 cm, Perimeter ≈ 25.28 cm
Final answer:

Rectangle: Area = 40 cm², Perimeter = 26 cm. Square with same area has perimeter ≈ 25.28 cm.

Applied rules:

• Rectangle Area: length × width

• Rectangle Perimeter: 2 × (length + width)

• Square Area: side², so side = √area

• Square Perimeter: 4 × side

⬜
Rectangle
8×5 cm
🟦
Square
Same area
2 Square and Triangle Application
Exercise 2
A square garden has a side length of 6 meters. A triangular flower bed with base 4 meters and height 3 meters is placed inside the garden. What is the area of the garden that is not covered by the flower bed?
Definition:

Composite Area: The area of a shape that includes multiple parts. Area not covered = Total area - Covered area.

Square area
6²=36 m²
Triangle area
(4×3)÷2=6 m²
Remaining area
36-6=30 m²
Step 1: Calculate square garden area

Area = side² = 6² = 36 m²

Step 2: Calculate triangle flower bed area

Area = (base × height) ÷ 2 = (4 × 3) ÷ 2 = 6 m²

Step 3: Calculate uncovered area

Uncovered area = Total area - Covered area = 36 - 6 = 30 m²

Step 4: Verify the calculation

30 + 6 = 36 ✓

30 m² uncovered
Final answer:

The area of the garden not covered by the flower bed is 30 m².

Applied rules:

• Square Area: side²

• Triangle Area: (base × height) ÷ 2

• Composite Areas: Total - Covered = Uncovered

⬜
Square
36 m²
6×6 m
🔺
Triangle
6 m²
4×3÷2
⬜-
Uncovered
30 m²
36-6
3 Perimeter and Area Comparison
Exercise 3
Compare the area and perimeter of two rectangles: Rectangle A (4 cm × 6 cm) and Rectangle B (3 cm × 8 cm). Which has a larger area? Which has a larger perimeter?
Definition:

Comparison: Calculating and comparing measurements of different shapes to identify similarities and differences.

Rect A
A=24, P=20
Rect B
A=24, P=22
Comparison
Same area, diff perimeter
Step 1: Calculate Rectangle A area and perimeter

Area = 4 × 6 = 24 cm², Perimeter = 2(4 + 6) = 20 cm

Step 2: Calculate Rectangle B area and perimeter

Area = 3 × 8 = 24 cm², Perimeter = 2(3 + 8) = 22 cm

Step 3: Compare areas

Rectangle A area = Rectangle B area = 24 cm² (same area)

Step 4: Compare perimeters

Rectangle A perimeter = 20 cm, Rectangle B perimeter = 22 cm

Step 5: Draw conclusion

Both have the same area, but Rectangle B has a larger perimeter.

Same area: 24 cm²
Different perimeters: 20 cm vs 22 cm
Final answer:

Both rectangles have the same area (24 cm²), but Rectangle B has a larger perimeter (22 cm vs 20 cm).

Applied rules:

• Rectangle Area: length × width

• Rectangle Perimeter: 2 × (length + width)

• Comparison: Same area can have different perimeters

Rectangle Dimensions Area Perimeter
A 4×6 cm 24 cm² 20 cm
B 3×8 cm 24 cm² 22 cm
Key Concepts: Area, Perimeter, and Shapes
Area = space inside
Perimeter = distance around
Square: A=s², P=4s
Rectangle: A=l×w, P=2(l+w)
Basic Formulas
Square
A=s², P=4s
All sides equal
Rectangle
A=l×w, P=2(l+w)
Opposite sides equal
Triangle
A=(b×h)÷2
Three sides
Key Definitions:

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

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

Square: A quadrilateral with all sides equal and all angles 90°

Rectangle: A quadrilateral with opposite sides equal and all angles 90°

Triangle: A three-sided polygon

Measurement Strategy:
  1. Identify Shape: Determine the type of shape
  2. Select Formula: Choose the appropriate area or perimeter formula
  3. Substitute Values: Replace variables with given measurements
  4. Calculate: Perform the arithmetic operations
  5. Include Units: Express answer in proper measurement units
Tip 1: Area is always in square units (cm², m², ft²), perimeter is in linear units (cm, m, ft).
Tip 2: Draw and label shapes to visualize problems.
Tip 3: Two shapes can have the same area but different perimeters.
Common Error: Mixing up area and perimeter formulas or forgetting to include units.
Memory Aid: "Area covers the floor, perimeter walks the fence."
Essential Rules:

• Square Area: A = s²

• Square Perimeter: P = 4s

• Rectangle Area: A = l × w

• Rectangle Perimeter: P = 2(l + w)

• Triangle Area: A = (b × h) ÷ 2

Square
A = s²
P = 4s
Rectangle
A = l × w
P = 2(l + w)
Triangle
A = (b × h) ÷ 2
P = a + b + c
📏
Perimeter
Distance around
🟥
Area
Space inside
Solution: Exercises 4 to 5
4 Composite Shape Problem
Exercise 4
A playground consists of a rectangular area (12 m × 8 m) with a square sandbox (3 m × 3 m) inside it. What is the area of the playground that is not the sandbox? What is the total perimeter of the playground?
Definition:

Composite Shape: A shape made up of multiple simpler shapes. Area of composite shape = Sum of individual areas.

Playground
12×8=96 m²
Sandbox
3²=9 m²
Usable area
96-9=87 m²
Step 1: Calculate playground area

Area = 12 m × 8 m = 96 m²

Step 2: Calculate sandbox area

Area = 3 m × 3 m = 9 m²

Step 3: Calculate usable area

Usable area = Playground area - Sandbox area = 96 - 9 = 87 m²

Step 4: Calculate playground perimeter

Perimeter = 2(12 + 8) = 2 × 20 = 40 m

Usable area: 87 m²
Playground perimeter: 40 m
Final answer:

The area of the playground that is not the sandbox is 87 m². The perimeter of the playground is 40 m.

Applied rules:

• Rectangle Area: length × width

• Square Area: side²

• Composite Areas: Subtract covered area from total

• Rectangle Perimeter: 2 × (length + width)

Playground
96 m²
Sandbox
9 m²
Usable
87 m²
Perimeter
40 m
5 Real-World Application
Exercise 5
Mrs. Johnson wants to carpet her living room (5 m × 4 m) and put a border around the edges. Carpet costs $8 per square meter and border costs $3 per meter. How much will the carpet cost? How much will the border cost?
Definition:

Real-World Application: Using area and perimeter concepts to solve practical problems involving cost calculations.

Area
5×4=20 m²
Perimeter
2(5+4)=18 m
Carpet cost
20×$8=$160
Border cost
18×$3=$54
Step 1: Calculate area for carpet

Area = 5 m × 4 m = 20 m²

Step 2: Calculate perimeter for border

Perimeter = 2(5 + 4) = 2 × 9 = 18 m

Step 3: Calculate carpet cost

Carpet cost = 20 m² × $8/m² = $160

Step 4: Calculate border cost

Border cost = 18 m × $3/m = $54

Step 5: Calculate total cost

Total cost = $160 + $54 = $214

Carpet cost: $160
Border cost: $54
Total: $214
Final answer:

The carpet will cost $160 and the border will cost $54. Total cost is $214.

Applied rules:

• Area Calculation: for surface coverage needs

• Perimeter Calculation: for edge materials

• Cost Calculation: area/perimeter × unit cost

🟥
Area
20 m²
Carpet
📏
Perimeter
18 m
Border
💰
Cost
$214
Total
Complete Guide: Area, Perimeter, and Shapes Review
Area: square units
Perimeter: linear units
A = length × width
P = 2(length + width)
Key Formulas
Key definitions:

Area: The measure of the space inside a two-dimensional shape, expressed in square units (cm², m², ft², etc.)

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

Square: A quadrilateral with all four sides equal in length and all angles measuring 90°

Rectangle: A quadrilateral with opposite sides equal and all angles measuring 90°

Triangle: A three-sided polygon with three angles

Complete measurement methodology:
  1. Identify Shape Type: Determine if square, rectangle, triangle, or composite
  2. Select Appropriate Formula: Match formula to the shape
  3. Substitute Known Values: Replace variables with measurements
  4. Perform Calculations: Carry out arithmetic operations
  5. Express in Correct Units: Include appropriate measurement units
  6. Verify Reasonableness: Check if answer makes sense
Tip 1: Area is "inside" (think "area" sounds like "interior"), Perimeter is "around" (think "rim").
Tip 2: Area is always in square units, perimeter is always in linear units.
Tip 3: Draw and label shapes to visualize the problem.
Tip 4: Always check that your answer has the correct units.
Common errors: Confusing area and perimeter formulas, forgetting to include units, mixing up length and width, calculating incorrectly.
Exam preparation: Practice with various shapes, memorize formulas, work on word problems, verify answers carefully.
Essential formulas and rules:

• Square Area: A = s² (side squared)

• Square Perimeter: P = 4s (4 times the side)

• Rectangle Area: A = l × w (length times width)

• Rectangle Perimeter: P = 2(l + w) (twice the sum of length and width)

• Triangle Area: A = (b × h) ÷ 2 (base times height divided by 2)

Shape Area Formula Perimeter Formula Example
Square s² 4s Side = 5: A=25, P=20
Rectangle l × w 2(l + w) 5×3: A=15, P=16
Triangle (b × h) ÷ 2 a + b + c b=4,h=3: A=6
💡
Strategy
Identify shape first
✅
Verification
Check units and logic
📚
Practice
Work with various shapes
Area
Square units
Inside space
Perimeter
Linear units
Edge distance
Units
cm, m, ft
cm², m², ft²

Questions & Answers

Question: My students mix up area and perimeter. How can I help them remember the difference?

Answer: Use memorable analogies:

  • Area: "If you were to tile the floor, how many tiles would you need?" (covers inside)
  • Perimeter: "If you were to walk around the edge, how far would you walk?" (goes around)
  • Memory trick: "Area" sounds like "interior", "Perimeter" sounds like "rim"
  • Hands-on: Use grid paper for area, string for perimeter

Practice with real objects to make the concepts tangible!

Question: My child sometimes forgets the formulas. Do you have any tips to help them remember?

Answer: Create memory aids:

  • Square: "Area = side squared, Perimeter = 4 sides"
  • Rectangle: "Area = length × width, Perimeter = 2 length + 2 width"
  • Practice: Draw shapes and write formulas repeatedly
  • Visual: Draw the shape and label what each formula measures
  • Units: Remember area is square units, perimeter is linear units

Practice regularly with flashcards or worksheets!

Question: I sometimes get confused about which units to use. Can you explain?

Answer: Think about what you're measuring:

  • Area: Measures space inside a shape - always square units (cm², m², ft²)
  • Perimeter: Measures distance around - always linear units (cm, m, ft)
  • Memory aid: Area = squares fit inside, Perimeter = measuring tape around
  • Example: A 3×4 rectangle has area 12 cm² and perimeter 14 cm

Always include the correct units in your final answer!