Mathematics Class 10 - Surface Areas-And-Volumes Notes

Surface Areas and Volumes

In this chapter, you will learn how to find the surface area and volume of different 3D shapes like cubes, cuboids, cylinders, cones, and spheres. By the end of this chapter, you will be able to use formulas to solve real-life and geometry problems involving these shapes.

Key Concepts

• Surface Area: The total area of all the faces or curved surfaces of a 3D shape.
• Volume: The amount of space occupied by a 3D shape.
• 3D Shapes: Objects that have length, breadth, and height (or radius and height).

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

Applications

• Finding the amount of material needed to make a box, can, or tank.
• Calculating the capacity of containers.
• Solving real-life problems involving packing, storing, and constructing objects.

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 cube has a side of 6 cm. Find its total surface area and volume.
4. Find the volume of a cone with base radius 3.5 cm and height 12 cm. (Use π = 22/7)
5. What is the surface area and volume of a sphere with radius 5 cm?

Challenge Yourself

• A cylindrical tank has a height of 2 m and a diameter of 1.4 m. Find its capacity in litres. (1 m³ = 1000 litres)
• If the total surface area of a cube is 150 cm², what is the length of its side?
• A hemisphere has a volume of 288π cm³. Find its radius.

Did You Know?

• The surface area of a sphere is the smallest among all shapes with the same volume!
• Cylinders are used in making cans, pipes, and tanks.

Glossary

• Surface Area: The total area covering the outside of a 3D shape.
• Volume: The space inside a 3D shape.
• π (Pi): A special number used in circles, approximately 3.14 or 22/7.
• Curved Surface Area: The area of the curved part of a 3D shape (not including the base or top).

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 = 2 × 22/7 × 7 × 10 = 2 × 22 × 10 = 440 cm²
   Volume = 22/7 × 7 × 7 × 10 = 22 × 7 × 10 = 1540 cm³
3. Total Surface Area = 6 × 6 × 6 = 6 × 36 = 216 cm²
   Volume = 6 × 6 × 6 = 216 cm³
4. Volume = (1/3) × 22/7 × 3.5 × 3.5 × 12 = (1/3) × 22/7 × 12.25 × 12 = (1/3) × 22/7 × 147 = (1/3) × 462 = 154 cm³
5. Surface Area = 4 × 22/7 × 5 × 5 = 4 × 22 × 25 / 7 = 2200 / 7 ≈ 314.29 cm²
   Volume = (4/3) × 22/7 × 5 × 5 × 5 = (4/3) × 22 × 125 / 7 = (4/3) × 2750 / 7 ≈ 523.81 cm³

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Practice using these formulas to solve real-life and geometry problems with 3D shapes!