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:

$\boxed{\text{Perimeter}=2\times (l+b)}$

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:

$\boxed{\text{Perimeter}=4\times s}$

Perimeter of an Equilateral Triangle

$\boxed{\text{Perimeter}=3\times \text{side}}$

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

$\boxed{\text{Area}=l\times b}$

For a rectangle 5 m long and 4 m wide: Area = 5 × 4 = 20 square metres (sq m).

Area of a Square

$\boxed{\text{Area}=s\times s=s^2}$

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):

8. Key Concepts Summary

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.