In this chapter, you will learn
- —Understand the concepts of perimeter and area
- —Calculate perimeter and area of rectangles and squares
- —Find area of triangles using base and height
- —Understand circles and calculate their circumference and area
- —Apply formulas to solve real-world problems
What is Perimeter and Area?
Perimeter is the total distance around the boundary of a 2D shape. We measure it in linear units (cm, m, etc.).
Area is the amount of space enclosed within a 2D shape. We measure it in square units (cm², m², etc.).
Key Difference:
Perimeter = Boundary length (add all sides)
Area = Space inside (depends on shape)
Real-Life Examples:
- Fencing a garden: Perimeter (length of fence needed)
- Painting a wall: Area (amount of paint needed)
- Running around a track: Perimeter (total distance)
- Tiling a floor: Area (number of tiles needed)
Exam Tip
Always write units clearly. Perimeter has 1 unit (cm, m), area has 2 units (cm², m²).
Common Mistake
Confusing perimeter and area. Remember: perimeter is the boundary, area is the inside.
Perimeter and Area of Rectangles
A rectangle has 4 sides with opposite sides equal and all angles 90°.
Formulas:
Perimeter of Rectangle = 2(l + b) where l = length, b = breadth
Area of Rectangle = l × b
Examples:
- Rectangle with l = 8 cm, b = 5 cm
- Perimeter = 2(8 + 5) = 2(13) = 26 cm
- Area = 8 × 5 = 40 cm²
Square (Special Rectangle): When l = b
- Perimeter of Square = 4a (where a = side)
- Area of Square = a²
- Example: Side = 5 cm → Perimeter = 20 cm, Area = 25 cm²
Exam Tip
For rectangles, make sure to add the length AND breadth before multiplying by 2.
Common Mistake
Using 2l + 2b instead of 2(l + b) - they're same but mistakes happen in calculation.
Area of Triangles
The area of a triangle depends on its base and height.
Formula:
Area of Triangle = (1/2) × base × height
Or: Area = (b × h) / 2
Important: Height must be perpendicular (⊥) to the base.
Example:
- Base = 6 cm, Height = 4 cm
- Area = (1/2) × 6 × 4 = (1/2) × 24 = 12 cm²
Types of Triangles:
- Right Triangle: Use two perpendicular sides as base and height
- Equilateral/Isosceles: Draw perpendicular from vertex to base
- Scalene: Use any side as base, draw height perpendicular to it
Exam Tip
Height must always be perpendicular to the base. Use the symbol ⊥ to show this.
Common Mistake
Using a slant side as height instead of the perpendicular height. This gives wrong area.
Circumference and Area of Circles
A circle is a 2D shape with all points equidistant from the center.
Key Terms:
- Radius (r): Distance from center to any point on circle
- Diameter (d): Distance across circle through center = 2r
- Circumference (C): Perimeter of circle = 2πr or πd
Formulas:
Circumference = 2πr = πd
Area = πr²
Where π ≈ 22/7 or 3.14
Example: Circle with radius = 7 cm
- Circumference = 2πr = 2 × (22/7) × 7 = 2 × 22 = 44 cm
- Area = πr² = (22/7) × 7² = (22/7) × 49 = 154 cm²
Exam Tip
Use π = 22/7 for exact answers. Use π = 3.14 only when asked or for approximations.
Common Mistake
Using diameter instead of radius in area formula. Area = πr², not π(d)².
Practical Applications and Problem-Solving
Area and perimeter are used in many real-world situations:
Example 1: Fencing a Garden
- A rectangular garden is 10 m × 6 m. Find cost to fence it at Rs. 50 per meter.
- Perimeter = 2(10 + 6) = 32 m
- Cost = 32 × 50 = Rs. 1600
Example 2: Painting a Wall
- Wall is 5 m × 3 m. Paint costs Rs. 100 per m². Find total cost.
- Area = 5 × 3 = 15 m²
- Cost = 15 × 100 = Rs. 1500
Example 3: Comparing Shapes
- A square of side 4 cm and rectangle 6 cm × 2 cm have same perimeter (16 cm)
- But area differs: Square = 16 cm², Rectangle = 12 cm²
- Same perimeter ≠ Same area
Exam Tip
Read questions carefully. Identify if you need perimeter or area. Check units in final answer.
Common Mistake
Not converting units properly (e.g., mixing m and cm) before calculating.
Chapter Summary
Area and Perimeter are fundamental measurements in geometry:
- Perimeter: Total distance around boundary (linear units)
- Area: Space enclosed inside (square units)
- Rectangle: P = 2(l+b), A = l×b
- Square: P = 4a, A = a²
- Triangle: A = (1/2) × base × height
- Circle: C = 2πr, A = πr²
Exam Focus: Formula application, word problems, unit conversions, comparing shapes.