Lecture on Calculating Surface Area and Volume of 3D Figures

In this lecture, we learned how to find the surface area and volume of various three-dimensional figures including cones, pyramids, prisms, cylinders, and spheres. The instructor emphasized understanding and memorizing key formulas and grouping similar shapes to simplify the learning process.

Prisms and Cylinders

• Definition: Prisms have two parallel and congruent bases; cylinders can be thought of as circular prisms.
• Volume Formula:
  • [\text{Volume} = \text{Area of the Base} \times \text{Height} (B \times h)]
  • Example: For a base area of 16 (\text{in}^2) and height 10 inches, volume is 160 (\text{in}^3).
• Surface Area Formula:
  • [\text{Surface Area} = 2B + P \times h]
  • Example: Calculating surface area of a square prism with perimeter 16 and height 10 gives 192 (\text{in}^2).

Triangular Prisms

• Volume:
  • Base is triangular: [\text{Volume} = \frac{1}{2}bh \times h\text{ (height of prism)}]
  • Example: Triangle base area calculation with base 3, height 4, prism height 12 results in 72 (\text{units}^3).
• Surface Area:
  • Formula: [\text{Surface Area} = 2B + P \times h]
  • Example: Triangular base with sides 3, 4, 5; height 12 results in 156 (\text{units}^2).

Cylinders

• Volume:
  • [\text{Volume} = \pi r^2 \times h]
  • Example: Radius 3, height 6 gives 54(\pi \text{ m}^3).
• Surface Area:
  • [\text{Surface Area} = 2\pi r^2 + 2\pi r \times h]
  • Example: Same radius and height yield 54(\pi \text{ m}^2).

Pyramids and Cones

• Grouping: Both have a singular base and a vertex.
• Volume Formula:
  • [\text{Volume} = \frac{1}{3}B \times h]
  • Example: Square pyramid with base 6, height 4 results in 48 (\text{in}^3).
• Surface Area Formula:
  • [\text{Surface Area} = B + \frac{1}{2}PL]
  • Example: Square pyramid with perimeter 24, slant height 5 results in 96 (\text{in}^2).
• Cone:
  • Same volume formula, but for circular base.
  • Example: Radius 5, height 12 results in 100(\pi \text{ units}^3).
  • Surface area for cone: Similar adaptation using circular base.

Spheres

• Volume Formula:
  • [\text{Volume} = \frac{4}{3}\pi r^3]
  • Example: Radius 3 results in 36(\pi \text{ units}^3).
• Surface Area Formula:
  • [\text{Surface Area} = 4\pi r^2]
  • Example: Results again in 36(\pi \text{ units}^2).

Additional Resources

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