Volume of Pyramids and Cones

Volume of a Pyramid

• To find the volume of a pyramid:
  • Find the area of the base (square, rectangle, triangle, hexagon, etc.).
  • Multiply by the perpendicular distance (altitude) of the pyramid.
  • Take one-third of the result.
  • Formula: $\frac{1}{3} \times \text{Area of Base} \times \text{Height}$
• If you don't take one-third, you get the volume of the whole prism.

Volume of a Cone

• A cone is the rounded version of a pyramid.
• The volume is calculated similarly: $\frac{1}{3} \times \text{Area of Base} \times \text{Altitude}$
• The base is a circle, so the area of the base is $πr^2$.
• Therefore, the volume of a cone is:$\frac{1}{3} \pi r^2 h$
• If you don't take one-third, you'll have the volume of the entire cylinder.

Using the Slant Height

• If you're given the slant height instead of the actual height:
  • Use the Pythagorean theorem to find the altitude (h) of the cone.
  • $r^2 + h^2 = \text{slant height}^2$

Example Problem

• Cone with radius (r) = 10 units and slant height = 16 units.
  • Find the volume.

Solution

1. Find the area of the circular base:
   • $A = \pi r^2 = \pi (10)^2 = 100\pi$
2. Find the altitude (h) using the Pythagorean theorem:
   • $h^2 + 10^2 = 16^2$
   • $h^2 + 100 = 256$
   • $h^2 = 156$
   • $h = \sqrt{156}$
3. Simplify the square root:
   • $156 = 4 \times 39$
   • $h = \sqrt{4 \times 39} = 2\sqrt{39}$
4. Calculate the volume:
   • $V = \frac{1}{3} \times \text{Area of Base} \times \text{Height}$
   • $V = \frac{1}{3} \times 100\pi \times 2\sqrt{39}$
   • $V = \frac{200\pi \sqrt{39}}{3}$

• The volume of the cone is $\frac{200\pi \sqrt{39}}{3}$ cubic units.