Class 9 Notes

Class 9 Mathematics Notes Chapter 13 (Surface Areas and Volumes) – Mathematics Book

Philoid

Philoid

— 5 min read

Mathematics
Alright class, let's focus on Chapter 13: Surface Areas and Volumes. This is a crucial chapter, not just for your Class 9 exams, but it forms the foundation for many quantitative aptitude sections in government exams. We'll break down the key concepts and formulas for each 3D shape you need to know. Pay close attention to the formulas and the distinction between Lateral/Curved Surface Area (LSA/CSA) and Total Surface Area (TSA).

Chapter 13: Surface Areas and Volumes - Detailed Notes

Key Concepts:

  • Surface Area: The total area of the surface of a three-dimensional object.
    • Lateral Surface Area (LSA) / Curved Surface Area (CSA): The area of the faces excluding the top and bottom faces (for shapes like cuboids, cubes, cylinders) or the area of the curved part (for cones, spheres, hemispheres).
    • Total Surface Area (TSA): The sum of the areas of all the surfaces of the 3D object.
  • Volume: The measure of the amount of space occupied by a three-dimensional object. It's measured in cubic units (like cm³, m³).

1. Cuboid

  • Description: A 3D shape with six rectangular faces. Think of a matchbox or a duster.
  • Dimensions: Length (l), Breadth (b), Height (h).
  • Formulas:
    • Lateral Surface Area (LSA): Area of the four walls = 2h(l + b)
    • Total Surface Area (TSA): Area of all six faces = 2(lb + bh + hl)
    • Volume (V): Space occupied = l × b × h
    • Length of Diagonal: √(l² + b² + h²) (Useful for competitive exams)

2. Cube

  • Description: A special type of cuboid where all edges are equal. All six faces are squares. Think of a dice.
  • Dimensions: Edge or Side (a). (So, l = b = h = a)
  • Formulas:
    • Lateral Surface Area (LSA): Area of four walls = 4a²
    • Total Surface Area (TSA): Area of all six faces = 6a²
    • Volume (V): Space occupied = a³
    • Length of Diagonal: a√3

3. Right Circular Cylinder

  • Description: A solid generated by revolving a rectangle about one of its sides. It has two circular bases (top and bottom) and a curved surface. Think of a pipe or a can. 'Right Circular' means the axis is perpendicular to the circular bases.
  • Dimensions: Radius of the circular base (r), Height (h).
  • Formulas:
    • Curved Surface Area (CSA): Area of the curved part = 2πrh
    • Total Surface Area (TSA): CSA + Area of two circular bases = 2πrh + 2πr² = 2πr(h + r)
    • Volume (V): Area of base × height = πr²h

4. Right Circular Cone

  • Description: A solid generated by revolving a right-angled triangle about one of its sides containing the right angle. It has one circular base, one vertex, and a curved surface. Think of an ice-cream cone or a conical tent.
  • Dimensions: Radius of the circular base (r), Height (h), Slant Height (l).
  • Important Relationship: The height, radius, and slant height form a right-angled triangle, with the slant height as the hypotenuse. So, l² = r² + h² or l = √(r² + h²).
  • Formulas:
    • Curved Surface Area (CSA): Area of the curved part = πrl
    • Total Surface Area (TSA): CSA + Area of circular base = πrl + πr² = πr(l + r)
    • Volume (V): (1/3) × Volume of cylinder with same base radius and height = (1/3)πr²h

5. Sphere

  • Description: A perfectly round geometrical object in three-dimensional space. All points on the surface are equidistant from the center. Think of a ball.
  • Dimensions: Radius (r).
  • Formulas:
    • Surface Area (SA): There's only one surface, so LSA = TSA = 4πr²
    • Volume (V): (4/3)πr³

6. Hemisphere

  • Description: Exactly half of a sphere, cut by a plane passing through the center. Think of a bowl.
  • Dimensions: Radius (r).
  • Formulas:
    • Curved Surface Area (CSA): Half the surface area of the sphere = 2πr²
    • Total Surface Area (TSA): CSA + Area of the circular top = 2πr² + πr² = 3πr²
    • Volume (V): Half the volume of the sphere = (2/3)πr³

Important Points for Exams:

  • Units: Always pay attention to units. Area is in square units (cm², m²) and Volume is in cubic units (cm³, m³). Ensure consistency in units before applying formulas. Convert if necessary (e.g., cm to m).
  • π Value: Use π = 22/7 or 3.14 as specified in the question. If not specified, 22/7 is generally preferred unless the radius/diameter is a multiple of 3.5 or 7.
  • Visualization: Try to visualize the shapes. Understand what LSA/CSA and TSA physically represent.
  • Formula Memorization: These formulas are fundamental. Write them down, revise them regularly.
  • Application: Practice problems involving finding surface area/volume, finding dimensions when area/volume is given, comparing volumes/areas of different shapes, and problems involving melting and recasting (where volume remains constant).

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. Choose the correct option.

  1. The lateral surface area of a cuboid with dimensions 10 cm × 8 cm × 5 cm is:
    a) 400 cm²
    b) 360 cm²
    c) 180 cm²
    d) 200 cm²

  2. If the volume of a cube is 729 cm³, its total surface area is:
    a) 486 cm²
    b) 324 cm²
    c) 729 cm²
    d) 81 cm²

  3. The curved surface area of a right circular cylinder of height 14 cm and base radius 3 cm is: (Use π = 22/7)
    a) 264 cm²
    b) 132 cm²
    c) 396 cm²
    d) 528 cm²

  4. The volume of a cone with radius 7 cm and height 6 cm is: (Use π = 22/7)
    a) 924 cm³
    b) 308 cm³
    c) 616 cm³
    d) 462 cm³

  5. The slant height of a cone is 10 cm and its base radius is 6 cm. Its height is:
    a) 4 cm
    b) 8 cm
    c) √136 cm
    d) 64 cm

  6. The surface area of a sphere with a diameter of 14 cm is: (Use π = 22/7)
    a) 308 cm²
    b) 1232 cm²
    c) 616 cm²
    d) 154 cm²

  7. The total surface area of a hemisphere of radius 'r' is:
    a) 2πr²
    b) 3πr²
    c) 4πr²
    d) (2/3)πr³

  8. If the radius of a sphere is doubled, its volume becomes:
    a) Double
    b) Four times
    c) Six times
    d) Eight times

  9. How many cubes of side 2 cm can be cut from a cuboid measuring 10 cm × 8 cm × 4 cm?
    a) 20
    b) 40
    c) 60
    d) 80

  10. The ratio of the volume of a cone, a hemisphere, and a cylinder, all having the same radius and same height (equal to the radius), is:
    a) 1:2:3
    b) 3:2:1
    c) 1:3:2
    d) 2:3:1

Answer Key for MCQs:

  1. c) 180 cm² (LSA = 2h(l+b) = 25(10+8) = 10 * 18 = 180)
  2. a) 486 cm² (Volume = a³ = 729 => a = 9 cm. TSA = 6a² = 6 * 9² = 6 * 81 = 486)
  3. a) 264 cm² (CSA = 2πrh = 2 * (22/7) * 3 * 14 = 2 * 22 * 3 * 2 = 264)
  4. b) 308 cm³ (Volume = (1/3)πr²h = (1/3) * (22/7) * 7² * 6 = (1/3) * 22 * 7 * 6 = 22 * 7 * 2 = 308)
  5. b) 8 cm (l² = r² + h² => 10² = 6² + h² => 100 = 36 + h² => h² = 64 => h = 8)
  6. c) 616 cm² (Diameter = 14 cm => Radius r = 7 cm. SA = 4πr² = 4 * (22/7) * 7² = 4 * 22 * 7 = 616)
  7. b) 3πr² (Direct formula)
  8. d) Eight times (V = (4/3)πr³. If r becomes 2r, New V = (4/3)π(2r)³ = (4/3)π(8r³) = 8 * [(4/3)πr³] = 8 * Original V)
  9. b) 40 (Volume of cuboid = 1084 = 320 cm³. Volume of small cube = 2³ = 8 cm³. Number of cubes = Vol_cuboid / Vol_cube = 320 / 8 = 40)
  10. a) 1:2:3 (Given h=r. V_cone = (1/3)πr²h = (1/3)πr³. V_hemisphere = (2/3)πr³. V_cylinder = πr²h = πr³. Ratio = (1/3)πr³ : (2/3)πr³ : πr³ = (1/3) : (2/3) : 1 = 1 : 2 : 3)

Make sure you understand the reasoning behind each answer. Practice more problems from your textbook and reference materials to gain mastery over this chapter. Good luck!

Read more