Mensuration — Class 6 Mathematics
'How much fence do you need? How much paint? Mensuration is the mathematics of measuring boundaries and surfaces.'
1. Introduction
Mensuration deals with measuring geometric figures — the length of the boundary (perimeter) and the space inside (area).
Why This Chapter Matters
- Used in construction, farming, interior design
- Fencing a field, tiling a floor, framing a picture
- AP context: measuring agricultural fields, constructing houses
Perimeter vs Area — The Big Difference
| Concept | Definition | Unit | Analogy |
|---|---|---|---|
| Perimeter | Distance AROUND a shape | Unit of length (cm, m, km) | Fencing a garden |
| Area | Space INSIDE a shape | Square units (cm², m²) | Grass covering the garden |
2. Perimeter
The perimeter is the total length of the boundary of a closed figure.
Finding Perimeter
Simply add up the lengths of ALL sides.
Example: A triangle with sides 5 cm, 7 cm, 9 cm Perimeter = 5 + 7 + 9 = 21 cm
3. Perimeter of a Rectangle
A rectangle has opposite sides equal.
l (length)
┌──────────┐
b │ │ b (breadth)
└──────────┘
l
Formula: Perimeter = 2 × (length + breadth) = 2(l + b)
Why This Formula?
Rectangle has 2 lengths and 2 breadths. Perimeter = l + b + l + b = 2l + 2b = 2(l + b)
Worked Examples
Example 1: A rectangle has length 12 cm and breadth 8 cm. Find its perimeter. P = 2(12 + 8) = 2 × 20 = 40 cm
Example 2 (AP Context): A farmer's rectangular field in Guntur is 50 m long and 30 m wide. How much fencing is needed for one round? P = 2(50 + 30) = 2 × 80 = 160 m
Example 3 (Finding missing side): The perimeter of a rectangle is 60 cm and its length is 18 cm. Find the breadth. P = 2(l + b) 60 = 2(18 + b) 30 = 18 + b b = 12 cm
4. Perimeter of a Square
A square has ALL four sides equal.
┌──────┐
│ │
s │ │ s (side)
│ │
└──────┘
s
Formula: Perimeter = 4 × side = 4s
Worked Examples
Example 1: A square has side 15 cm. Find its perimeter. P = 4 × 15 = 60 cm
Example 2: The perimeter of a square is 88 m. Find its side. 4s = 88 s = 22 m
Example 3: A square park has side 120 m. Ravi walks 3 rounds around it. How much distance does he cover? Perimeter = 4 × 120 = 480 m 3 rounds = 3 × 480 = 1440 m
5. Perimeter of an Equilateral Triangle
An equilateral triangle has ALL three sides equal.
/\
/ \
/ \
/______\
s (side)
Formula: Perimeter = 3 × side = 3s
Worked Example
An equilateral triangle has side 9 cm. Find its perimeter. P = 3 × 9 = 27 cm
6. Perimeter of a Regular Polygon
A regular polygon has ALL sides equal AND all angles equal.
Formula: Perimeter = Number of sides × Length of each side
| Polygon | Number of Sides (n) | Perimeter |
|---|---|---|
| Regular Pentagon | 5 | 5 × side |
| Regular Hexagon | 6 | 6 × side |
| Regular Octagon | 8 | 8 × side |
| Regular n-gon | n | n × side |
Worked Example
A regular hexagon has side 8 cm. Find its perimeter. P = 6 × 8 = 48 cm
7. Area
The area is the amount of surface enclosed by a closed figure.
Unit of Area
- Small areas: square centimetre (cm²) — area of a square of side 1 cm
- Large areas: square metre (m²) — area of a square of side 1 m
- Very large: square kilometre (km²), hectare (1 ha = 10,000 m²)
8. Area of a Rectangle
Formula: Area = length × breadth = l × b
Why Length × Breadth?
A rectangle of length l and breadth b can be divided into l × b unit squares.
Worked Examples
Example 1: A rectangle has length 15 cm and breadth 10 cm. Find its area. A = 15 × 10 = 150 cm²
Example 2 (AP Context): A classroom in a Vijayawada school is 12 m long and 8 m wide. What is its area? How many students can sit if each needs 2 m²? Area = 12 × 8 = 96 m² Students = 96 ÷ 2 = 48 students
Example 3 (Finding missing side): The area of a rectangle is 120 m² and its length is 15 m. Find the breadth. A = l × b 120 = 15 × b b = 120 ÷ 15 = 8 m
9. Area of a Square
Formula: Area = side × side = s²
Worked Examples
Example 1: A square has side 9 cm. Find its area. A = 9 × 9 = 81 cm²
Example 2: A square park has side 25 m. Find its area. A = 25 × 25 = 625 m²
Example 3: Find the side of a square whose area is 144 cm². s² = 144 s = √144 = 12 cm
10. Area vs Perimeter — Detailed Comparison
Different Units
| Aspect | Perimeter | Area |
|---|---|---|
| Unit | cm, m, km | cm², m², km² |
| What it measures | Boundary length | Surface enclosed |
Different Shapes, Same Perimeter (or Area)
It is possible for two different shapes to have the same perimeter but different areas, or the same area but different perimeters.
Example: Compare these rectangles:
| Rectangle | Length | Breadth | Perimeter | Area |
|---|---|---|---|---|
| A | 8 cm | 2 cm | 20 cm | 16 cm² |
| B | 6 cm | 4 cm | 20 cm | 24 cm² |
| C | 5 cm | 5 cm | 20 cm | 25 cm² |
Same perimeter (20 cm) but different areas! Among all rectangles with a given perimeter, the square has the maximum area.
Real-Life Examples
| Situation | Perimeter or Area? |
|---|---|
| Fencing a field | Perimeter |
| Painting a wall | Area |
| Laying tiles on floor | Area |
| Putting a frame around a photo | Perimeter |
| Planting grass in a garden | Area |
| Border around a saree | Perimeter |
11. Word Problems
Problem 1 (Fencing — Perimeter)
A rectangular field in Kurnool measures 80 m by 45 m. The farmer wants to fence it with 4 rounds of wire. How much wire is needed?
Solution: Perimeter = 2(80 + 45) = 2 × 125 = 250 m 4 rounds = 4 × 250 = 1000 m Answer: 1000 m of wire
Problem 2 (Tiling — Area)
A hall is 15 m long and 10 m wide. How many square tiles of side 50 cm are needed to tile the floor?
Solution: Area of hall = 15 × 10 = 150 m² = 150 × 10,000 = 15,00,000 cm² Area of one tile = 50 × 50 = 2500 cm² Number of tiles = 15,00,000 ÷ 2500 = 600 tiles Answer: 600 tiles
Problem 3 (Both)
A square park of side 100 m has a path of width 2 m inside along its boundary. Find the area of the path and the cost of fencing the park at Rs. 25 per metre.
Solution: Part 1 — Area of path: Outer square side = 100 m, Outer area = 100 × 100 = 10,000 m² Inner square side = 100 − 4 = 96 m, Inner area = 96 × 96 = 9216 m² Area of path = 10,000 − 9216 = 784 m²
Part 2 — Fencing cost: Perimeter of park = 4 × 100 = 400 m Cost = 400 × 25 = Rs. 10,000
Problem 4 (AP Agriculture)
A farmer in Krishna district has two fields:
- Field A: Square of side 60 m
- Field B: Rectangle of length 80 m and breadth 45 m
Which field has larger area? Which needs more fencing?
Solution: Area A = 60 × 60 = 3600 m² Area B = 80 × 45 = 3600 m² Areas are EQUAL.
Perimeter A = 4 × 60 = 240 m Perimeter B = 2(80 + 45) = 250 m Field B needs more fencing even though areas are equal!
12. Common Mistakes — Fix Them Now
| # | Mistake | Correction |
|---|---|---|
| 1 | Confusing perimeter and area units | Perimeter: cm (not cm²). Area: cm² (not cm) |
| 2 | Adding only 2 sides for rectangle perimeter | Rectangle has 4 sides: 2(l+b), not l+b |
| 3 | Forgetting unit conversion | 1 m = 100 cm, but 1 m² = 10,000 cm² (100 × 100) |
| 4 | Using wrong formula for square area | Area = side², not 4×side (that's perimeter) |
| 5 | Not adding all sides for irregular shapes | Add EVERY side of the boundary |
13. Exam Focus
Marks Blueprint
| Question Type | Marks | Topic |
|---|---|---|
| MCQ | 1 | Formula / Unit identification |
| Short answer | 2 | Find perimeter of rectangle/square |
| Short answer | 2 | Find area of rectangle/square |
| Word problem | 3 | Real-life perimeter or area |
| Long problem | 4 | Combined perimeter and area problem |
Quick Self-Test (5 Questions)
Q1: Find the perimeter of a rectangle with l = 14 cm, b = 11 cm.
<details><summary>Answer</summary>P = 2(14+11) = 2×25 = 50 cm</details>Q2: What is the area of a square with side 13 cm?
<details><summary>Answer</summary>A = 13×13 = 169 cm²</details>Q3: A rectangle has perimeter 48 cm and length 16 cm. Find its breadth.
<details><summary>Answer</summary>48 = 2(16+b) → 24 = 16+b → b = 8 cm</details>Q4: A carpet of size 5 m × 4 m costs Rs. 200 per m². What is the total cost?
<details><summary>Answer</summary>Area = 20 m². Cost = 20×200 = Rs. 4000</details>Q5: Which has larger area: a square of side 12 m or a rectangle of 14 m × 10 m?
<details><summary>Answer</summary>Square area = 144 m², Rectangle area = 140 m². Square is larger.</details>14. Chapter Summary
- Perimeter: distance around a figure (units: cm, m, km)
- Rectangle: P = 2(l + b)
- Square: P = 4s
- Equilateral triangle: P = 3s
- Regular polygon: P = n × s
- Area: space inside a figure (units: cm², m², km²)
- Rectangle: A = l × b
- Square: A = s²
- Key distinction: perimeter = boundary length, area = surface enclosed
- Same perimeter ≠ same area (square gives maximum area)
'Mensuration connects mathematics to the real world — from the classroom floor to the farmer's field.'
