Mathematics Class 11 - Surface Areas-And-Volumes Notes

Surface Areas and Volumes

Welcome to the chapter on Surface Areas and Volumes for Class 11. In this chapter, you will learn how to find the surface area and volume of different 3D shapes, understand their properties, and solve real-life problems using formulas. By the end of this chapter, you will be able to apply these concepts to geometry and practical situations.

Key Concepts

• Surface Area: The total area that the surface of a 3D object covers.
• Volume: The amount of space occupied by a 3D object.
• 3D Shapes: Solid figures like cuboids, cubes, cylinders, cones, spheres, and hemispheres.

Surface Area and Volume Formulas

• Cuboid:
  • Total Surface Area = 2(lb + bh + hl)
  • Volume = l × b × h
• Cube:
  • Total Surface Area = 6a²
  • Volume = a³
• Cylinder:
  • Curved Surface Area = 2πrh
  • Total Surface Area = 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

Properties and Applications

• Surface area helps in finding the amount of material needed to cover a solid.
• Volume helps in finding the capacity or space inside a solid.
• Used in real-life situations like painting, packaging, storing liquids, and construction.

Practice Questions

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

Challenge Yourself

• If the total surface area of a cube is 150 cm², find the length of its side.
• A cylindrical tank has a height of 2 m and a diameter of 1.4 m. How much water can it hold?
• Compare the volumes of a sphere and a hemisphere with the same radius.

Did You Know?

• The surface area of a sphere is the minimum possible for a given volume among all solids.
• The volume of air in a football is calculated using the formula for the volume of a sphere.

Glossary

• Surface Area: The total area of the outer surface of a solid.
• Volume: The amount of space inside a solid.
• Curved Surface Area: The area of the curved part of a solid (like the side of a cylinder or cone).
• π (Pi): A mathematical constant, approximately 3.14.

Answers to Practice Questions

1. Total Surface Area = 2(10×5 + 5×4 + 4×10) = 2(50 + 20 + 40) = 2×110 = 220 cm²
   Volume = 10 × 5 × 4 = 200 cm³
2. Curved Surface Area = 2πrh = 2 × 3.14 × 7 × 10 = 439.6 cm²
   Volume = πr²h = 3.14 × 7 × 7 × 10 = 1538.6 cm³
3. Volume = (1/3)πr²h = (1/3) × 3.14 × 3 × 3 × 4 = (1/3) × 3.14 × 9 × 4 = (1/3) × 3.14 × 36 = 37.68 cm³
4. Surface Area = 4πr² = 4 × 3.14 × 5 × 5 = 314 cm²
5. Volume = (2/3)πr³ = (2/3) × 3.14 × 6 × 6 × 6 = (2/3) × 3.14 × 216 = (2/3) × 678.24 = 452.16 cm³

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Practice finding surface areas and volumes to master geometry and solve real-world problems!