Surface Areas and Volumes

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.