Chapter 11: Surface Areas and Volumes

Summary

• Curved surface area of a cone: πrl
• Total surface area of a right circular cone: πrl + πr² = πr(l + r)
• Surface area of a sphere (radius r): 4πr²
• Curved surface area of a hemisphere: 2πr²
• Total surface area of a hemisphere: 3πr²
• Volume of a cone: (1/3)πr²h
• Volume of a sphere (radius r): (4/3)πr³
• Volume of a hemisphere: (2/3)πr³

Key Formulas and Definitions

Learning Objectives

• Calculate the surface area of cones and spheres.
• Determine the volume of cones, spheres, and hemispheres.
• Apply the formulas for surface area and volume in practical problems.
• Understand the relationship between the dimensions of cones and their surface areas.

Common Mistakes and Exam Tips

• Mistake: Confusing the formulas for surface area and volume. Tip: Always write down the formula before substituting values.
• Mistake: Forgetting to convert units when necessary. Tip: Check units in the problem statement and convert them if needed.
• Mistake: Incorrectly calculating the slant height in cone problems. Tip: Use the Pythagorean theorem to find the slant height accurately.

Important Diagrams

• Diagram of a Sphere: Shows a sphere divided into two hemispheres, illustrating the concept of surface area.
• Diagram of Cone and Cylinder Setup: Demonstrates the relationship between the volume of a cone and a cylinder, showing that three cones fill one cylinder.
• Diagram of Volume Measurement: Illustrates the method of measuring the volume of a sphere through water displacement.

Chapter 11: Surface Areas and Volumes

11.1 Surface Area of a Right Circular Cone

• Curved Surface Area:
  • Formula: πrl
  • Where:
    • r = base radius
    • l = slant height
• Total Surface Area:
  • Formula: πrl + πr² = πr(l + r)

11.2 Surface Area of a Sphere

• Surface Area:
  • Formula: 4πr²
  • Where: r = radius of the sphere

11.3 Volume of a Right Circular Cone

• Volume:
  • Formula: (1/3)πr²h
  • Where:
    • r = base radius
    • h = height

11.4 Volume of a Sphere

• Volume:
  • Formula: (4/3)πr³
  • Where: r = radius of the sphere
• Volume of a Hemisphere:
  • Formula: (2/3)πr³

Summary of Key Formulas

1. Curved Surface Area of a Cone: πrl
2. Total Surface Area of a Cone: πr(l + r)
3. Surface Area of a Sphere: 4πr²
4. Volume of a Cone: (1/3)πr²h
5. Volume of a Sphere: (4/3)πr³
6. Volume of a Hemisphere: (2/3)πr³

Examples

• Example 1: Find the curved surface area of a right circular cone with slant height 10 cm and base radius 7 cm.
  • Solution: Curved Surface Area = πrl = 22/7 * 7 * 10 cm² = 220 cm²
• Example 2: Find the volume of a sphere of radius 11.2 cm.
  • Solution: Volume = (4/3)π(11.2)³ cm³ = 5887.32 cm³ (approx)

Exercises

1. Find the volume of a sphere whose radius is:
   • (i) 7 cm
   • (ii) 0.63 m
2. Find the amount of water displaced by a solid spherical ball of diameter:
   • (i) 28 cm
   • (ii) 0.21 m
3. The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g/cm³?
4. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
5. How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold?
6. A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.
7. Find the volume of a sphere whose surface area is 154 cm².
8. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹ 4989.60. If the cost of white-washing is ₹ 20 per square metre, find:
   • (i) inside surface area of the dome
   • (ii) volume of the air inside the dome.
9. Twenty-seven solid iron spheres, each of radius r and surface area S, are melted to form a sphere with surface area S'. Find:
   • (i) radius r' of the new sphere
   • (ii) ratio of S and S'.
10. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm³) is needed to fill this capsule?

Common Mistakes and Exam Tips

Common Pitfalls

• Misunderstanding Volume Formulas: Students often confuse the volume formulas for spheres, cones, and hemispheres. Ensure you memorize the correct formulas:
  • Volume of a Sphere: $V = \frac{4}{3} \pi r^3$
  • Volume of a Cone: $V = \frac{1}{3} \pi r^2 h$
  • Volume of a Hemisphere: $V = \frac{2}{3} \pi r^3$
• Incorrect Unit Conversions: Be careful with unit conversions, especially when dealing with measurements in cm and m. Always convert to the same unit before performing calculations.
• Neglecting to Use π Correctly: Some students forget to use the value of π correctly. In exercises, it is often given as $\pi = \frac{22}{7}$ or 3.14. Make sure to apply the correct value as specified.
• Forgetting to Include All Parts of Surface Area: When calculating the total surface area of cones or hemispheres, remember to include both the curved surface area and the base area if applicable.

Exam Tips

• Practice with Different Shapes: Work on problems involving various shapes (spheres, cones, hemispheres) to become familiar with their properties and formulas.
• Draw Diagrams: Whenever possible, draw diagrams to visualize the problem. This can help in understanding the relationships between different dimensions.
• Check Your Work: After solving a problem, take a moment to review your calculations and ensure that you have used the correct formulas and units.
• Time Management: Allocate your time wisely during exams. If you find a problem too challenging, move on and return to it later if time permits.