Class 11 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 various three-dimensional (3D) shapes. Understanding these concepts is essential for solving real-life problems involving space, capacity, and covering surfaces.

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 such as cuboids, cubes, cylinders, cones, spheres, and hemispheres.

Formulas for Common Solids

  • 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
  • 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 container (surface area).
  • Calculating the capacity of tanks, boxes, and other containers (volume).
  • Solving real-life problems involving painting, wrapping, or filling objects.

Sample Problems

  1. Find the surface area and volume of a cube with side 5 cm.
    Solution:
    Surface Area = 6 × 52 = 150 cm2
    Volume = 53 = 125 cm3
  2. A cylindrical tank has radius 3 m and height 7 m. Find its total surface area and volume.
    Solution:
    TSA = 2π × 3 × (3 + 7) = 2π × 3 × 10 = 60π ≈ 188.4 m2
    Volume = π × 32 × 7 = π × 9 × 7 = 63π ≈ 197.9 m3
  3. Find the volume of a cone with radius 4 cm and height 9 cm.
    Solution:
    Volume = (1/3)π × 42 × 9 = (1/3)π × 16 × 9 = (1/3)π × 144 = 48π ≈ 150.8 cm3

Practice Exercises

  1. Find the total surface area and volume of a cuboid of length 8 cm, breadth 5 cm, and height 3 cm.
  2. A sphere has a radius of 7 cm. Calculate its surface area and volume.
  3. What is the curved surface area of a cylinder with radius 2.5 cm and height 10 cm?
  4. Find the volume of a hemisphere with radius 6 cm.
  5. A cone has a slant height of 10 cm and base radius 6 cm. Find its total surface area.

Summary

  • Surface area measures the covering of a 3D object; volume measures the space inside it.
  • Each solid shape has specific formulas for surface area and volume.
  • These concepts are widely used in real-life situations and higher mathematics.