Volume Calculations: Pyramids and Cones

Volume of a Pyramid

• Formula: $V = \frac{1}{3}Ah$, where:
  • $A$ = Area of the base
  • $h$ = Perpendicular height
• The base can be any shape; use the appropriate area formula.
• If the height isn't directly given, Pythagoras' theorem may be needed to find the perpendicular height.

Example: Pyramid Volume Calculation

1. Identify the base area
   • If the base is a square with side $s$, then $A = s^2$
2. Substitute values into the volume formula
   • $V = \frac{1}{3} Ah$
3. Calculations
   *If $A = 16 m^2$ and $h = 6 m$, then $V = \frac{1}{3} * 16 * 6 = 32 m^3$

Volume of a Cone

• Formula: $V = \frac{1}{3}πr^2h$
  • $r$ = Radius of the circular base
  • $h$ = Perpendicular height
• The perpendicular height is needed for volume, not the slant height.

Volume vs Surface Area of Cone

• Volume and surface area are different:
  • Volume uses perpendicular height $h$.
  • Surface area uses slant height $l$ in the formula $SA = πr^2 + πrl$.

Example: Cone Volume Calculation

1. Identify the radius and height
   • Make sure to halve the diameter to get the radius.
   • Ensure the height is perpendicular.
2. Substitute values into the volume formula
   • $V = \frac{1}{3} π r^2 h$

Capacity and Volume

• 1 cubic centimeter (cm^3) = 1 milliliter (mL)
• 1 cubic meter = 1 kiloliter
• Converting volume to capacity is straightforward with cubic centimeters to milliliters.

Finding Height or Radius Using the Volume of a Cone

1. Write down the formula: $V = \frac{1}{3}πr^2h$
2. Substitute known values
3. Rearrange and solve

Finding Slant Height

• Use Pythagoras' theorem if needed: $c^2 = a^2 + b^2$ to find the slant height ($l$) given the radius ($r$) and height ($h$).