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
|
Shape |
Formula |
Example |
|
Square |
P=4×side |
Side = 5 cm → P = 4 × 5 = 20 cm |
|
Rectangle |
P=2(l+b) |
l = 8 cm, b = 4 cm → P = 2(8+4) = 24 cm |
|
Triangle |
P=a+b+c |
a = 3 cm, b = 4 cm, c = 5 cm → P = 3+4+5 = 12 cm |
|
Circle (Circumference) |
C=2πr |
r = 7 cm → C ≈ 2 × 3.14 × 7 = 44 cm |
📌 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
|
Shape |
Formula |
Example |
|
Square |
A=side^2 |
Side = 6 cm → A = 6² = 36 cm² |
|
Rectangle |
A=l×b |
l = 7 cm, b = 4 cm → A = 7 × 4 = 28 cm² |
|
Triangle |
A=1/2×(b×h) |
b = 10 cm, h = 6 cm → A = ½ × 10 × 6 = 30 cm² |
|
Circle |
A=πr^2 |
r = 5 cm → A ≈ 3.14 × 5² = 78.5 cm² |
📌 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)
- What does perimeter measure?
A) Space inside
B) Boundary length
C) Volume
D) Height - Area measures:
A) Boundary
B) Space inside
C) Length only
D) Height - Perimeter of a square =
A) side²
B) 2 × side
C) 4 × side
D) side³ - Area of a square =
A) 2 × side
B) side²
C) 4 × side
D) side³ - Perimeter of a rectangle =
A) l × b
B) 2(l + b)
C) l²
D) b² - Area of rectangle =
A) l + b
B) 2(l + b)
C) l × b
D) l² - Unit of perimeter is:
A) cm²
B) cm
C) m²
D) none - Unit of area is:
A) cm
B) m
C) cm²
D) km - Perimeter of triangle =
A) a + b + c
B) a × b × c
C) a²
D) 2a - Area of triangle =
A) b × h
B) ½ × b × h
C) b + h
D) h² - Perimeter of square with side 6 cm =
A) 12 cm
B) 24 cm
C) 36 cm
D) 18 cm - Area of square with side 5 cm =
A) 10 cm²
B) 20 cm²
C) 25 cm²
D) 30 cm² - Perimeter of rectangle (l=8, b=4) =
A) 12 cm
B) 24 cm
C) 32 cm
D) 16 cm - Area of rectangle (7 × 3) =
A) 21 cm²
B) 14 cm²
C) 10 cm²
D) 24 cm² - Perimeter of triangle (3, 4, 5) =
A) 10
B) 11
C) 12
D) 13 - Area of triangle (b=10, h=6) =
A) 60
B) 30
C) 20
D) 50 - Circumference of circle formula =
A) πr²
B) 2πr
C) r²
D) πr - Area of circle formula =
A) 2πr
B) πr²
C) r²
D) πd - Circumference when r=7 cm =
A) 44 cm
B) 49 cm
C) 22 cm
D) 14 cm - Area when r=5 cm =
A) 78.5
B) 50
C) 25
D) 100 - If side doubles, area becomes:
A) Double
B) Half
C) Four times
D) Same - If side doubles, perimeter becomes:
A) Double
B) Four times
C) Half
D) Same - Area of square = 64 cm², side =
A) 6
B) 7
C) 8
D) 9 - Perimeter of square with side 9 cm =
A) 18
B) 36
C) 27
D) 45 - Area of rectangle (l=10, b=2) =
A) 12
B) 20
C) 24
D) 30 - Fence length required =
A) Area
B) Perimeter
C) Volume
D) Height - Paint needed depends on:
A) Perimeter
B) Area
C) Length
D) Width - Tiles required depends on:
A) Area
B) Perimeter
C) Height
D) Length - Running track distance =
A) Area
B) Perimeter
C) Height
D) Radius - Carpet needed for room =
A) Area
B) Perimeter
C) Length
D) Height - Boundary of garden =
A) Area
B) Perimeter
C) Height
D) Volume - Floor covering requires:
A) Perimeter
B) Area
C) Length
D) Width - Painting wall uses:
A) Perimeter
B) Area
C) Length
D) Height - Wire needed for boundary =
A) Area
B) Perimeter
C) Volume
D) Height - Book cover surface =
A) Area
B) Perimeter
C) Volume
D) Height - Shape with equal sides =
A) Rectangle
B) Square
C) Triangle
D) Circle - Formula for triangle area depends on:
A) Side only
B) Base & height
C) Radius
D) Diameter - Circle has:
A) Sides
B) Corners
C) Radius
D) Length - π value approx =
A) 2.14
B) 3.14
C) 4.14
D) 1.14 - Diameter =
A) r/2
B) 2r
C) r²
D) πr - Area of square side 12 =
A) 144
B) 24
C) 36
D) 48 - Perimeter of rectangle (5,3) =
A) 15
B) 16
C) 14
D) 12 - Area of triangle (b=8, h=5) =
A) 40
B) 20
C) 30
D) 25 - Circumference if diameter 14 =
A) 44
B) 22
C) 28
D) 14 - Area increases when:
A) Side increases
B) Side decreases
C) Nothing changes
D) Divide side - Perimeter depends on:
A) Boundary
B) Space
C) Volume
D) Height - Area depends on:
A) Boundary
B) Space inside
C) Height only
D) Length only - Composite shapes are:
A) Single shapes
B) Combined shapes
C) Only circles
D) Only squares - To solve complex shapes:
A) Ignore
B) Break into parts
C) Guess
D) Multiply - 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
