PERIMETER & AREA

PERIMETER & AREA: THE MATHEMATICS OF SPACE

📖 Introduction: Why Do We Need Perimeter and Area?

Imagine you want to fence your garden, paint your house walls, or buy a carpet for your room. You need to measure how much material is required. Perimeter helps in measuring the boundary, while area tells us how much space is covered.

📌 Memory Trick:
✔ Perimeter = Path around (fencing, boundary)
✔ Area = Space inside (paint, carpet, land)

🛤️ What is Perimeter?

📌 Definition: Perimeter is the total length of the boundary of a closed shape.

📌 Formula:

Perimeter=Sum of all sides\text{Perimeter} = \text{Sum of all sides}Perimeter=Sum of all sides

🔸 Perimeter Formulas of Common Shapes

📌 Shortcut for Regular Shapes:
✔ If all sides are equal, just multiply by the number of sides!

🛋️ What is Area?

📌 Definition: Area is the amount of surface covered by a shape.

📌 Formula:

Area=Space inside the boundary\text{Area} = \text{Space inside the boundary}Area=Space inside the boundary

🔹 Area Formulas of Common Shapes

📌 Quick Trick:
✔ Square = side × side
✔ Rectangle = length × breadth
✔ Triangle = ½ × base × height
✔ Circle = π × radius²

🏡 Real-Life Applications of Perimeter and Area

✅ Garden Fence: To find the fence length, calculate perimeter.
✅ Painting Walls: To find how much paint is needed, calculate area.
✅ Buying Tiles or Carpets: To cover a floor, measure area.
✅ Running Tracks: To know how much you run, find the perimeter of the track.

🚀 How to Find Perimeter and Area of Complex Shapes?

1️⃣ Composite Figures (Combination of Shapes)

Example: Find the area of an L-shaped figure
1️⃣ Divide it into smaller rectangles.
2️⃣ Find the area of each rectangle.
3️⃣ Add them together to get the total area.

2️⃣ Finding the Shaded Region

Example: A circular field with a square garden inside
1️⃣ Find the area of the circle.
2️⃣ Find the area of the square.
3️⃣ Subtract square’s area from circle’s area to get the remaining shaded part.

📌 Trick:
“Break complex shapes into smaller known shapes!”

📊 Word Problems (With Solutions!)

🔹 Example 1: Find the Perimeter of a Triangle

A triangle has sides 5 cm, 7 cm, and 9 cm. Find its perimeter.
✔ Solution:

P=5+7+9=21 cm

🔹 Example 2: Find the Area of a Circle

A circular park has a radius of 10 m. Find its area.
✔ Solution:

A=πr^2=3.14×10^2=314 m^2

🔥 Smart Tricks to Remember Formulas!

📌 Story Method:
Imagine you have a chocolate bar 🍫
✔ If you measure the wrapper’s border, it’s perimeter.
✔ If you measure the whole chocolate inside, it’s area.

📌 Shortcut for Exams:
✔ Square: Multiply side × side
✔ Rectangle: Multiply length × breadth
✔ Triangle: Half of base × height
✔ Circle:
➖ Circumference: 2πr
➖ Area: πr^2

🚀 Challenge Questions!

1️⃣ A rectangular garden is 15 m long and 10 m wide. Find:
✔ Perimeter
✔ Area

2️⃣ A circular pizza has a radius of 7 cm. Find:
✔ Circumference
✔ Area

3️⃣ A square park has a side of 20 m. A path of 2 m width runs along the border. Find:
✔ The area of the park
✔ The area of the path

🔑 Summary & Key Takeaways

✔ Perimeter = Total boundary length
✔ Area = Total surface covered
✔ Use formulas for different shapes
✔ Break complex shapes into smaller ones
✔ Real-life applications: Fencing, painting, flooring, running tracks!

PERIMETER & AREA – 50 MCQs

🔹 Questions (1–50)

1. What does perimeter measure?
   A) Space inside
   B) Boundary length
   C) Volume
   D) Height
2. Area measures:
   A) Boundary
   B) Space inside
   C) Length only
   D) Height
3. Perimeter of a square =
   A) side²
   B) 2 × side
   C) 4 × side
   D) side³
4. Area of a square =
   A) 2 × side
   B) side²
   C) 4 × side
   D) side³
5. Perimeter of a rectangle =
   A) l × b
   B) 2(l + b)
   C) l²
   D) b²
6. Area of rectangle =
   A) l + b
   B) 2(l + b)
   C) l × b
   D) l²
7. Unit of perimeter is:
   A) cm²
   B) cm
   C) m²
   D) none
8. Unit of area is:
   A) cm
   B) m
   C) cm²
   D) km
9. Perimeter of triangle =
   A) a + b + c
   B) a × b × c
   C) a²
   D) 2a
10. Area of triangle =
    A) b × h
    B) ½ × b × h
    C) b + h
    D) h²
11. Perimeter of square with side 6 cm =
    A) 12 cm
    B) 24 cm
    C) 36 cm
    D) 18 cm
12. Area of square with side 5 cm =
    A) 10 cm²
    B) 20 cm²
    C) 25 cm²
    D) 30 cm²
13. Perimeter of rectangle (l=8, b=4) =
    A) 12 cm
    B) 24 cm
    C) 32 cm
    D) 16 cm
14. Area of rectangle (7 × 3) =
    A) 21 cm²
    B) 14 cm²
    C) 10 cm²
    D) 24 cm²
15. Perimeter of triangle (3, 4, 5) =
    A) 10
    B) 11
    C) 12
    D) 13
16. Area of triangle (b=10, h=6) =
    A) 60
    B) 30
    C) 20
    D) 50
17. Circumference of circle formula =
    A) πr²
    B) 2πr
    C) r²
    D) πr
18. Area of circle formula =
    A) 2πr
    B) πr²
    C) r²
    D) πd
19. Circumference when r=7 cm =
    A) 44 cm
    B) 49 cm
    C) 22 cm
    D) 14 cm
20. Area when r=5 cm =
    A) 78.5
    B) 50
    C) 25
    D) 100
21. If side doubles, area becomes:
    A) Double
    B) Half
    C) Four times
    D) Same
22. If side doubles, perimeter becomes:
    A) Double
    B) Four times
    C) Half
    D) Same
23. Area of square = 64 cm², side =
    A) 6
    B) 7
    C) 8
    D) 9
24. Perimeter of square with side 9 cm =
    A) 18
    B) 36
    C) 27
    D) 45
25. Area of rectangle (l=10, b=2) =
    A) 12
    B) 20
    C) 24
    D) 30
26. Fence length required =
    A) Area
    B) Perimeter
    C) Volume
    D) Height
27. Paint needed depends on:
    A) Perimeter
    B) Area
    C) Length
    D) Width
28. Tiles required depends on:
    A) Area
    B) Perimeter
    C) Height
    D) Length
29. Running track distance =
    A) Area
    B) Perimeter
    C) Height
    D) Radius
30. Carpet needed for room =
    A) Area
    B) Perimeter
    C) Length
    D) Height
31. Boundary of garden =
    A) Area
    B) Perimeter
    C) Height
    D) Volume
32. Floor covering requires:
    A) Perimeter
    B) Area
    C) Length
    D) Width
33. Painting wall uses:
    A) Perimeter
    B) Area
    C) Length
    D) Height
34. Wire needed for boundary =
    A) Area
    B) Perimeter
    C) Volume
    D) Height
35. Book cover surface =
    A) Area
    B) Perimeter
    C) Volume
    D) Height
36. Shape with equal sides =
    A) Rectangle
    B) Square
    C) Triangle
    D) Circle
37. Formula for triangle area depends on:
    A) Side only
    B) Base & height
    C) Radius
    D) Diameter
38. Circle has:
    A) Sides
    B) Corners
    C) Radius
    D) Length
39. π value approx =
    A) 2.14
    B) 3.14
    C) 4.14
    D) 1.14
40. Diameter =
    A) r/2
    B) 2r
    C) r²
    D) πr
41. Area of square side 12 =
    A) 144
    B) 24
    C) 36
    D) 48
42. Perimeter of rectangle (5,3) =
    A) 15
    B) 16
    C) 14
    D) 12
43. Area of triangle (b=8, h=5) =
    A) 40
    B) 20
    C) 30
    D) 25
44. Circumference if diameter 14 =
    A) 44
    B) 22
    C) 28
    D) 14
45. Area increases when:
    A) Side increases
    B) Side decreases
    C) Nothing changes
    D) Divide side
46. Perimeter depends on:
    A) Boundary
    B) Space
    C) Volume
    D) Height
47. Area depends on:
    A) Boundary
    B) Space inside
    C) Height only
    D) Length only
48. Composite shapes are:
    A) Single shapes
    B) Combined shapes
    C) Only circles
    D) Only squares
49. To solve complex shapes:
    A) Ignore
    B) Break into parts
    C) Guess
    D) Multiply
50. Best way to remember formulas:
    A) Ignore
    B) Practice
    C) Memorize without use
    D) Skip

✅ ANSWER KEY

1-B, 2-B, 3-C, 4-B, 5-B,
6-C, 7-B, 8-C, 9-A, 10-B,
11-B, 12-C, 13-B, 14-A, 15-C,
16-B, 17-B, 18-B, 19-A, 20-A,
21-C, 22-A, 23-C, 24-B, 25-B,
26-B, 27-B, 28-A, 29-B, 30-A,
31-B, 32-B, 33-B, 34-B, 35-A,
36-B, 37-B, 38-C, 39-B, 40-B,
41-A, 42-B, 43-B, 44-A, 45-A,
46-A, 47-B, 48-B, 49-B, 50-B