Perimeter and Area — Class 6 Maths (Ganita Prakash)
1. About This Chapter
Perimeter and Area introduces two of the most practical measurement concepts in mathematics. The perimeter tells us about the boundary — the distance around a shape. The area tells us about the space inside — how much surface a shape covers. From fencing a garden to laying a carpet, from framing a picture to tiling a floor, these two concepts are used everywhere.
2. Perimeter — The Distance Around
The perimeter of a closed shape is the total length of its boundary. Imagine walking along the edge of a playground — the total distance you cover is the perimeter.
Perimeter of a Rectangle
For a rectangle of length l and breadth b:
Why? A rectangle has 2 lengths and 2 breadths. So perimeter = l + b + l + b = 2l + 2b = 2(l+b).
Perimeter of a Square
For a square of side s:
Perimeter of an Equilateral Triangle
Perimeter of Any Polygon
Add up the lengths of ALL sides. For regular polygons (all sides equal), perimeter = number of sides × length of one side.
3. Worked Perimeter Examples
Example 1: Rectangular tablecloth
A tablecloth is 3 m long and 2 m wide. Find its perimeter.
Solution: P = 2 × (3 + 2) = 2 × 5 = 10 metres.
Example 2: Square park
A square park has side 75 m. Find its perimeter.
Solution: P = 4 × 75 = 300 metres.
Example 3: Wire for fencing
How much wire is needed to fence a rectangular garden 20 m by 15 m with 3 rounds of wire?
Solution:
- Perimeter = 2 × (20 + 15) = 2 × 35 = 70 m
- Wire needed = 70 × 3 = 210 metres
4. Area — The Space Inside
While perimeter measures the boundary, area measures the amount of surface inside a closed shape. Area is measured in square units (sq cm, sq m, sq km).
Area of a Rectangle
For a rectangle 5 m long and 4 m wide: Area = 5 × 4 = 20 square metres (sq m).
Area of a Square
For a square of side 6 cm: Area = 6 × 6 = 36 sq cm.
5. Perimeter vs Area — They're NOT the Same!
A crucial insight: shapes with the same perimeter can have different areas, and shapes with the same area can have different perimeters.
Example: Same Perimeter, Different Area
- Rectangle A: l = 8, b = 2. Perimeter = 2(8+2) = 20. Area = 8×2 = 16.
- Rectangle B: l = 6, b = 4. Perimeter = 2(6+4) = 20. Area = 6×4 = 24.
Both have perimeter 20, but Rectangle B has MORE area (24 vs 16)!
Example: Same Area, Different Perimeter
- Square of side 4: Area = 16, Perimeter = 16.
- Rectangle 8×2: Area = 16, Perimeter = 20.
Same area, but the rectangle needs MORE fencing (20 vs 16)!
This concept is important for design efficiency — a square shape maximizes area for a given perimeter.
6. Real-Life Applications
Carpet on a Floor
A floor is 5 m long and 4 m wide. You place a square carpet of side 3 m on it.
- Floor area = 5 × 4 = 20 sq m
- Carpet area = 3 × 3 = 9 sq m
- Uncovered area = 20 − 9 = 11 sq m
Fencing a Garden
A rectangular garden 30 m × 20 m needs fencing. Cost is ₹50 per metre.
- Perimeter = 2(30+20) = 100 m
- Cost = 100 × ₹50 = ₹5,000
Tiling a Wall
A wall 4 m × 3 m is to be tiled with square tiles of side 25 cm.
- Wall area = 400 cm × 300 cm = 1,20,000 sq cm
- Tile area = 25 × 25 = 625 sq cm
- Number of tiles = 1,20,000 ÷ 625 = 192 tiles
7. Perimeter and Area of Regular Polygons
For regular polygons (all sides and angles equal):
| Polygon | Perimeter | Area Approach |
|---|---|---|
| Equilateral Triangle | 3 × side | Introduced conceptually |
| Square | 4 × side | side² |
| Regular Pentagon | 5 × side | Introduced later |
| Regular Hexagon | 6 × side | Introduced later |
8. Key Concepts Summary
| Concept | Formula | Example (l=5, b=3) |
|---|---|---|
| Perimeter (Rectangle) | 2(l + b) | 2(5+3) = 16 |
| Perimeter (Square) | 4s | 4×5 = 20 |
| Area (Rectangle) | l × b | 5×3 = 15 sq units |
| Area (Square) | s² | 5² = 25 sq units |
9. Important Vocabulary
- Perimeter: Total length of the boundary of a closed shape
- Area: Amount of surface enclosed within a shape, measured in square units
- Square Unit: Unit of area — sq cm (cm²), sq m (m²), sq km (km²)
- Regular Polygon: A polygon with all sides and all angles equal
- Dimension: Measurement of length, breadth, or side
10. Conclusion
Perimeter and Area bridges the abstract world of numbers with the physical world around us. These concepts answer very practical questions: How much fencing? How much paint? How many tiles? How much carpet? But beyond practical utility, the chapter teaches an important mathematical insight — that perimeter and area are independent properties. Understanding their relationship (and their differences) is critical for design, architecture, engineering, and everyday problem-solving.
