Class 10 Notes - Surface Areas and Volumes

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Surface Areas and Volumes

This chapter explores the formulas and concepts needed to calculate the surface areas and volumes of various three-dimensional (3D) shapes. Understanding these concepts helps in solving real-life problems involving containers, buildings, and other objects.

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 shapes have different formulas for surface area and volume.

Important 3D Shapes and Their Formulas

Shape Surface Area Volume
Cuboid 2(lb + bh + hl) l × b × h
Cube 6a2 a3
Cylinder 2πr(r + h) πr2h
Cone πr(l + r) (1/3)πr2h
Sphere 4πr2 (4/3)πr3
Hemisphere 3πr2 (curved + base) (2/3)πr3

Applications

  • Finding the amount of material needed to make a box, can, or tent.
  • Calculating the capacity of tanks, containers, and rooms.
  • Solving real-life problems involving cost, 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 = 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 = π × 72 × 3 = π × 49 × 3 = 462π m3 ≈ 1451.33 m3
  3. Find the total surface area of a sphere of radius 10 cm.
    Solution:
    Surface area = 4π × 102 = 4π × 100 = 400π cm2 ≈ 1256.64 cm2

Tips

  • Always use the same units for all dimensions.
  • Remember to include units in your answers (cm2 for area, cm3 for volume).
  • Draw diagrams to visualize the problem.
  • For composite shapes, break them into simpler parts.

Practice Exercises

  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 volume.
  4. What is the surface area of a hemisphere with radius 6 cm?
  5. How much water can a spherical tank of radius 2 m hold?

Summary

  • Surface area is the total area covering a 3D object.
  • Volume is the space occupied by a 3D object.
  • Different shapes have different formulas for surface area and volume.
  • Apply these concepts to solve real-world problems.