Class 6 Mathematics Notes Chapter 10 (Mensuration) – Mathematics Book

Alright students, let's dive into Chapter 10: Mensuration from your Class 6 NCERT Mathematics book. This is a crucial chapter, not just for your class exams, but also because these fundamental concepts frequently appear in various government exams. Mensuration is essentially about measuring geometric figures – specifically their boundaries and the space they occupy.

Chapter 10: Mensuration - Detailed Notes

1. Introduction

• Mensuration: The branch of geometry that deals with the measurement of length, area, and volume of plane and solid figures.
• In this chapter (Class 6 level), we focus on two key measurements for plane (2D) figures:
  • Perimeter: The distance around a closed figure.
  • Area: The measure of the surface enclosed by a closed figure.

2. Perimeter

• Definition: Perimeter is the total length of the boundary of a closed plane figure. Imagine walking along the edges of a field; the total distance you cover is its perimeter.

• Units: Perimeter is measured in units of length like centimetres (cm), metres (m), kilometres (km), etc.

• Perimeter of a Rectangle:

  • A rectangle has four sides, with opposite sides being equal in length. Let the length be 'l' and the breadth (or width) be 'b'.
  • Perimeter = Sum of all sides = l + b + l + b = 2l + 2b
  • Formula: Perimeter of a Rectangle = 2 × (length + breadth) = 2(l + b)
  • Example: A rectangular park is 50 m long and 30 m wide. Its perimeter = 2 × (50 + 30) = 2 × 80 = 160 m.

• Perimeter of a Square:

  • A square is a special rectangle where all four sides are equal. Let the length of a side be 's'.
  • Perimeter = Sum of all sides = s + s + s + s = 4s
  • Formula: Perimeter of a Square = 4 × side = 4s
  • Example: A square picture frame has a side of 25 cm. Its perimeter = 4 × 25 = 100 cm.

• Perimeter of an Equilateral Triangle:

  • An equilateral triangle has three equal sides. Let the length of a side be 'a'.
  • Perimeter = Sum of all sides = a + a + a = 3a
  • Formula: Perimeter of an Equilateral Triangle = 3 × side = 3a

• Perimeter of Regular Polygons:

  • A regular polygon is a closed figure with all sides equal and all angles equal (e.g., square, equilateral triangle, regular pentagon, regular hexagon).
  • General Formula: Perimeter of a Regular Polygon = Number of sides × Length of one side
  • Example: A regular pentagon (5 sides) has a side length of 8 cm. Its perimeter = 5 × 8 = 40 cm.
  • Example: A regular hexagon (6 sides) has a side length of 10 m. Its perimeter = 6 × 10 = 60 m.

• Perimeter of Irregular Shapes: For shapes with sides of different lengths, simply add the lengths of all the sides to find the perimeter.

3. Area

• Definition: Area is the amount of surface or region covered by a closed plane figure. Think of it as the space inside the boundary.

• Units: Area is measured in square units, like square centimetres (cm²), square metres (m²), square kilometres (km²), etc.

• Concept using Squared Paper: We can estimate the area of irregular shapes by placing them on a grid paper (graph paper) and counting the number of squares covered.

  • Count fully filled squares as 1 unit².
  • Count more than half-filled squares as 1 unit².
  • Count exactly half-filled squares as ½ unit².
  • Ignore less than half-filled squares.
  • The sum gives an approximate area.

• Area of a Rectangle:

  • The area of a rectangle is the product of its length and breadth.
  • Formula: Area of a Rectangle = length × breadth = l × b
  • Example: A rectangular garden is 12 m long and 5 m wide. Its area = 12 × 5 = 60 m².

• Area of a Square:

  • Since a square has equal sides (s), its area is the product of its side with itself.
  • Formula: Area of a Square = side × side = s²
  • Example: A square tile has a side of 15 cm. Its area = 15 × 15 = 225 cm².

4. Key Formulas Summary

5. Important Points for Government Exams

• Units are Crucial: Always pay attention to the units given in the question and the units required in the answer. Be prepared to convert units (e.g., cm to m, m to km, cm² to m²).
  • 1 m = 100 cm
  • 1 km = 1000 m
  • 1 m² = 1 m × 1 m = 100 cm × 100 cm = 10,000 cm²
• Distinguish Perimeter and Area: Understand the difference. Perimeter is length (1D), Area is surface (2D). Questions might involve finding the cost of fencing (perimeter) or the cost of tiling/carpeting (area).
• Word Problems: Practice applying the formulas to real-world scenarios described in word problems. Identify the shape, the given dimensions, and what needs to be calculated (perimeter or area).
• Finding Missing Dimensions: Sometimes, the perimeter or area might be given, and you'll need to find the length, breadth, or side. This involves rearranging the formulas.
  • If Perimeter (P) and length (l) of a rectangle are given, breadth b = (P/2) - l.
  • If Area (A) and length (l) of a rectangle are given, breadth b = A / l.
  • If Perimeter (P) of a square is given, side s = P / 4.
  • If Area (A) of a square is given, side s = √A (Square root concept might be slightly advanced for Class 6 but good to know).

Multiple Choice Questions (MCQs)

1. The perimeter of a rectangle with length 15 cm and breadth 10 cm is:
   a) 25 cm
   b) 50 cm
   c) 150 cm
   d) 100 cm

2. What is the area of a square park whose side is 60 m?
   a) 240 m²
   b) 3600 m
   c) 3600 m²
   d) 120 m²

3. A thin wire 80 cm long is formed into a square. What is the length of the side of the square?
   a) 4 cm
   b) 20 cm
   c) 16 cm
   d) 40 cm

4. The area of a rectangular sheet is 500 cm². If the length of the sheet is 25 cm, what is its width?
   a) 20 cm
   b) 10 cm
   c) 25 cm
   d) 12.5 cm

5. The perimeter of a regular pentagon is 100 cm. How long is its each side?
   a) 10 cm
   b) 25 cm
   c) 20 cm
   d) 5 cm

6. Find the cost of fencing a square park of side 50 m at the rate of ₹ 20 per metre.
   a) ₹ 1000
   b) ₹ 4000
   c) ₹ 50000
   d) ₹ 2000

7. What happens to the area of a square if its side is doubled?
   a) Area remains same
   b) Area is doubled
   c) Area becomes four times
   d) Area is halved

8. Two regular hexagons of perimeter 30 cm each are joined side by side as shown in the figure (imagine two hexagons joined along one edge). The perimeter of the new figure is:
   a) 60 cm
   b) 55 cm
   c) 50 cm
   d) 65 cm
   (Self-correction: Need to be careful here. A regular hexagon has 6 sides. Perimeter 30 cm means side = 30/6 = 5 cm. When joined, 2 sides are internal. Total sides = 6+6 = 12. Internal sides = 2. External sides = 12-2 = 10. Perimeter = 10 * 5 = 50 cm)

9. The space occupied by a flat shape is called its:
   a) Perimeter
   b) Volume
   c) Area
   d) Boundary

10. A rectangular floor is 5 m long and 4 m wide. How many square tiles with side 1 m are required to cover the floor?
    a) 9
    b) 18
    c) 20
    d) 25

Answer Key for MCQs:

1. b) 50 cm (Perimeter = 2(15+10) = 2(25) = 50)
2. c) 3600 m² (Area = 60 * 60 = 3600)
3. b) 20 cm (Side = Perimeter / 4 = 80 / 4 = 20)
4. a) 20 cm (Width = Area / Length = 500 / 25 = 20)
5. c) 20 cm (Side = Perimeter / Number of sides = 100 / 5 = 20)
6. b) ₹ 4000 (Perimeter = 4 * 50 = 200 m. Cost = 200 * 20 = 4000)
7. c) Area becomes four times (Original Area = s². New side = 2s. New Area = (2s)² = 4s²)
8. c) 50 cm (Side = 30/6 = 5 cm. Total sides = 12. Internal sides = 2. External sides = 10. Perimeter = 10 * 5 = 50)
9. c) Area
10. c) 20 (Area of floor = 5 * 4 = 20 m². Area of tile = 1 * 1 = 1 m². Number of tiles = Area of floor / Area of tile = 20 / 1 = 20)

Remember to practice the exercises in your NCERT book thoroughly. These concepts form the building blocks for more complex mensuration problems you'll encounter later. Good luck with your preparation!