Class 9 Notes - Surface Areas and Volumes

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Surface Areas and Volumes

In this chapter, you will learn how to calculate the surface areas and volumes of different 3D shapes such as cubes, cuboids, cylinders, cones, and spheres. Understanding these concepts helps in solving real-life problems related to capacity, wrapping, painting, and construction.

Key Concepts

  • Surface Area: The total area that the surface of a 3D object occupies.
  • Volume: The amount of space enclosed within a 3D object.
  • Different formulas are used for different shapes.

Formulas

  • Cuboid
    • Surface Area = 2(lb + bh + hl)
    • Volume = l × b × h
  • Cube
    • Surface Area = 6a2
    • Volume = a3
  • Cylinder
    • Curved Surface Area (CSA) = 2πrh
    • Total Surface Area (TSA) = 2πr(r + h)
    • Volume = πr2h
  • Cone
    • Curved Surface Area = πrl
    • Total Surface Area = πr(r + l)
    • Volume = (1/3)πr2h
    • l = slant height = √(r2 + h2)
  • Sphere
    • Surface Area = 4πr2
    • Volume = (4/3)πr3
  • Hemisphere
    • Curved Surface Area = 2πr2
    • Total Surface Area = 3πr2
    • Volume = (2/3)πr3

Applications

  • Finding the amount of material needed to make a box, can, or tent.
  • Calculating the capacity of tanks, bottles, or containers.
  • Estimating the cost of painting or wrapping objects.

Sample Problems

  1. Find the surface area and volume of a cube of side 5 cm.
    Solution:
    Surface Area = 6 × 52 = 6 × 25 = 150 cm2
    Volume = 53 = 125 cm3
  2. A cylindrical tank has radius 7 m and height 3 m. Find its volume.
    Solution:
    Volume = πr2h = (22/7) × 7 × 7 × 3 = 462 m3
  3. Find the total surface area of a sphere of radius 10 cm.
    Solution:
    Surface Area = 4πr2 = 4 × (22/7) × 10 × 10 = 1257.14 cm2

Practice Questions

  1. Find the volume of a cuboid with length 8 cm, breadth 5 cm, and height 2 cm.
  2. Calculate the curved surface area of a cylinder with radius 4 cm and height 10 cm.
  3. A cone has a base radius of 3 cm and height 4 cm. Find its slant height and total surface area.
  4. What is the volume of a hemisphere with radius 6 cm?
  5. How much canvas is required to make a tent in the shape of a cone with base radius 7 m and slant height 24 m?

Summary

  • Surface area measures the total area covering a 3D object.
  • Volume measures the space inside a 3D object.
  • Different shapes have different formulas for surface area and volume.
  • These concepts are useful in real-life situations like construction, packaging, and storage.