Lecture Notes: Calculating Surface Area and Volume of 3D Figures

Introduction

• Focus on how to find the surface area and volume of 3D figures: cones, pyramids, prisms, cylinders, and spheres.
• Learn easy ways to remember and group formulas for solving problems efficiently.

Prisms and Cylinders

Definitions

• Prism: A 3D figure with two parallel and congruent bases.
• Cylinder: Treated as a circular prism.

Volume Formula

• Formula: Volume = Area of Base ( B ) ( \times ) Height ( H )
• Example Calculation:
  • Square base with area 16 in², height 10 in.
  • Volume: ( 16 \times 10 = 160 ) in³.

Surface Area Formula

• Formula: Surface Area = 2B + Perimeter ( P ) ( \times ) Height ( H )
• Example Calculation:
  • Perimeter of square base = 16, height = 10.
  • Surface Area: ( 16 \times 10 + 32 = 192 ) in².

Triangular Prism

• Volume: Area of Base ( B ) ( \times ) Height ( H )
  • Base is a triangle: ( \frac{1}{2} \times 3 \times 4 \times 12 )
  • Total Volume = 72 units³
• Surface Area: Total outer area
  • Formula: ( 2B + PH )
  • Example Calculation gives 56 units²

Cylinder

• Similar to prism calculations.
• Volume: ( \pi r^2 \times H )
  • Example: ( \pi \times 3^2 \times 6 = 54\pi ) m³
• Surface Area: ( 2\pi r^2 + 2\pi rH )
  • Example: ( 54\pi ) m²

Pyramids and Cones

Definitions

• Pyramid: One base, tapers to a point.
• Cone: Circular base, tapers to a point.

Volume Formula

• Formula: ( \frac{1}{3} \times \text{Area of Base} \times \text{Height} )
• Pyramid: Example gives 48 in³.
• Cone: Example gives 100( \pi ) units³.

Surface Area Formula

• Formula: ( \text{Area of Base} + \frac{1}{2} \text{Perimeter} \times \text{Slant Height} )
• Pyramid: Example gives 96 in².
• Cone: Example gives 90( \pi ) units².

Spheres

Definition

• Sphere: All points equidistant from center.

Volume Formula

• Formula: ( \frac{4}{3}\pi r^3 )
• Example: 36( \pi ) units³

Surface Area Formula

• Formula: ( 4\pi r^2 )
• Example: 36( \pi ) units²

Conclusion

• Group shapes to simplify formula memorization.
• For further study, consider resources like SAT/ACT prep courses.
• Explore more math tutorials for additional help.