Surface Area and Volume of 3D Figures

Overview

• Learn to find the surface area and volume of various three-dimensional figures:
  • Cones
  • Pyramids
  • Prisms
  • Cylinders
  • Spheres
• Grouping formulas to ease memorization and problem-solving.

Prisms and Cylinders

Definition of Prisms

• A prism is a 3D figure with two parallel and congruent bases.
• The bases are separated by a height.

Grouping with Cylinders

• Cylinders are circular prisms (bases are circles).

Volume Formula

• Volume (V) = Area of the base (B) × Height (H)
  • Example:
    • Base area of a square = 16 in²
    • Height = 10 in
    • V = 16 × 10 = 160 in³

Surface Area Formula

• Surface Area (SA) = 2B + Perimeter of the base (P) × Height (H)
  • Example:
    • Area of the base (square) = 16 in² (2 bases = 32 in²)
    • Perimeter of the base = 16 in
    • Height = 10 in
    • SA = 32 + (16 × 10) = 192 in²

Triangular Prism Example

• Volume:

  • Area of triangle base = 1/2 × base × height = 1/2 × 3 × 4 = 6
  • Overall height = 12
  • V = 6 × 12 = 72 units³

• Surface Area:

  • SA = 2B + PH, where B = triangular area,
  • Total surface area = 144 + 12 = 156 units²

Cylinders

Volume of Cylinders

• Volume (V) = Area of the base (circle) × Height
  • Base area = πr²
  • Example:
    • Radius = 3, Height = 6
    • V = π × (3²) × 6 = 54π m³

Surface Area of Cylinders

• Surface Area (SA) = 2B + PH
  • Example:
    • Two bases: 2πr² (2 circles)
    • Perimeter = 2πr
    • SA = 54π m²

Pyramids and Cones

Definition

• Pyramids and cones both have one base.
• Pyramids have a square base; cones have a circular base.

Volume Formula

• Volume (V) = 1/3 × Area of the base (B) × Height (H)

  • Pyramid Example:
    • Base area (square) = 6 × 6 = 36
    • Overall height = 4
    • V = 1/3 × 36 × 4 = 48 in³

• Cone Example:

  • Base area = πr², with r = 5
  • V = 1/3 × π × 25 × 12 = 100π units³

Surface Area Formula

• Surface Area (SA) = Area of base + 1/2 × Perimeter × Slant height

  • Pyramid Example:
    • Base area = 36, Perimeter = 24, Slant height = 5
    • SA = 36 + 60 = 96 units²

• Cone Example:

  • SA = πr² + 1/2 × (2πr) × slant height
  • SA = 25π + 65π = 90π units²

Spheres

Definition

• A sphere is a set of all points equidistant from a center point.

Volume Formula

• Volume (V) = 4/3πr³
  • Example:
    • For radius r = 3, V = 4/3 × π × (3³) = 36π units³

Surface Area Formula

• Surface Area (SA) = 4πr²
  • Example:
    • For radius r = 3, SA = 4 × π × (3²) = 36π units²

Conclusion

• Group figures to simplify learning.
• Remember the key formulas for volume and surface area:
  • Prisms/Cylinders: V = B × H, SA = 2B + PH
  • Pyramids/Cones: V = 1/3 × B × H, SA = B + 1/2 × P × l
  • Spheres: V = 4/3πr³, SA = 4πr²
• Additional resources available for SAT/ACT prep.