Lecture Notes: Surface Area and Volume of Spheres

Formulas for Spheres

• Surface Area Formula:
  • ( 4 \pi r^2 )
  • Analogy: A baseball can be unraveled to form four circles (each with area ( \pi r^2 ), corresponding to the formula ( 4 \pi r^2 )).
• Volume Formula:
  • ( \frac{4}{3} \pi r^3 )

Example 1: Regular Sphere

• Given: Radius ( r = 27 ) millimeters
• Surface Area Calculation:
  • Formula: ( 4 \pi r^2 )
  • Calculation: ( 4 \pi (27^2) = 2916 \pi ) mm²
• Volume Calculation:
  • Formula: ( \frac{4}{3} \pi r^3 )
  • Calculation: ( \frac{4}{3} \times 19683 = 26244 \pi ) mm³

Solving for Radius from Volume

• Procedure:
  1. Set formula ( \frac{4}{3} \pi r^3 ) equal to a known value.
  2. Eliminate ( \pi ) and multiply by ( \frac{3}{4} ).
  3. Solve for ( r^3 ) and calculate the cube root.
  4. Example: Cube root of 4125 gives ( r \approx 16.038 ).

Example 2: Partial Sphere (with a Portion Cut Out)

• Surface Area Considerations:
  • Consider different parts: inside, curved part, sector.
  • Bottom part: ( 2 \pi r^2 )
  • Curved part: Fraction of ( 2 \pi r^2 ) with a certain angle.
  • Add half circles and sectors as needed.
• Volume Considerations:
  • Consider each hemisphere separately.
  • For top portion, consider fractional part of ( \frac{4}{3} \pi r^3 ) based on angle.

Detailed Calculations for Partial Sphere

• Surface Area:

  • Total calculation involves combining results from different sections: base (( 32\pi )), curve (( 29.33\pi )), other components.
  • Total: ( 70.67 \pi ) square feet.

• Volume:

  • Bottom half: ( \frac{2}{3} \times 64 \pi )
  • Top portion: Reduced to ( 39.11 \pi )
  • Total volume: ( 81.778 \pi ) cubic feet.

Key Points

• Surface area and volume of spheres require understanding of geometry formulas.
• Real-world analogies (e.g., baseball) can help visualize these concepts.
• Decomposing a sphere into parts helps in complex calculations like partial spheres.
• Use of correct units (mm² for area, mm³ for volume) is crucial.