Lecture Notes: Calculating Volume and Surface Area of a Cylinder

Key Concepts

• Cylinder Components:

  • Radius (r): The distance from the center to the edge of the base.
  • Height (h): The distance between the bases.
  • Base: The circular face of the cylinder.

• Formulas

  • Volume of a Cylinder:

    • Formula: ( V = \pi r^2 h )
    • Explanation: Volume is the area of the base (a circle) multiplied by the height.

  • Surface Area of a Cylinder:

    • Formula: ( SA = 2\pi r^2 + 2\pi rh )
    • Explanation: Surface area is the sum of the area of both circular bases and the lateral area.
    • Lateral Area (LA):
      • Formula: ( LA = 2\pi rh )
      • Explanation: Represents the area around the side of the cylinder, analogous to the paper wrapping a can.

Example Problems

Problem 1:

Calculate the Volume

• Given:
  • Radius = 5 cm
  • Height = 10 cm
• Solution:
  • ( V = \pi (5^2) \times 10 = 250\pi )
  • Exact answer: 250\pi
  • Approximate answer: 785.40 cubic cm

Problem 2:

Calculate the Surface Area

• Given:
  • Radius = 7 in
  • Height = 12 in
• Solution:
  • ( SA = 2\pi (7^2) + 2\pi (7)(12) )
  • ( = 98\pi + 168\pi = 266\pi )
  • Exact answer: 266\pi
  • Approximate answer: 835.66 square in

Problem 3:

Find Surface Area Given Volume

• Given:
  • Volume = 324\pi cubic cm
  • Height = 9 cm
• Solution:
  • Use volume formula to find radius: ( 324 = r^2 \times 9 \Rightarrow r = 6 )
  • Calculate Surface Area: ( SA = 2\pi (6^2) + 2\pi (6)(9) )
  • ( = 72\pi + 108\pi = 180\pi )
  • Approximate answer: 565.4 square cm

Problem 4:

Calculate Volume in Cubic Inches with Mixed Units

• Given:
  • Radius = 2.5 ft
  • Height = 14 in
• Convert Radius:
  • 1 ft = 12 in, so 2.5 ft = 30 in
• Solution:
  • ( V = \pi (30^2) \times 14 )
  • ( = 12,600\pi ) cubic in

Important Points

• Always ensure unit consistency when calculating, particularly when converting between units (e.g., feet to inches).
• Surface area measurements are in square units, whereas volume measurements are in cubic units.