Volume of a Cone

Key Concepts

• Volume Formula: Volume = Area of base × Height
  • For a cylinder:
    • Base = circle
    • Area of base = πr² (where r is the radius)
    • Height = vertical height of the cylinder
    • Volume = πr²h

Comparison between Cone and Cylinder

• Cone vs. Cylinder:
  • More water can fit in a cylinder than in a cone if both have the same radius and height.
  • The cone's volume is derived from the cylinder's volume.

Volume of a Cone Formula

• Mathematicians have established that for pyramid-shaped objects (including cones):
  • Volume of a cone = (Area of base × Height) / 3
  • This can also be expressed as:
    • Volume = (1/3) × Area of base × Height
• Symbols:
  • A = Area of base
  • h = Height
  • The fraction 1/3 signifies dividing by three.

Summary of Relationships

• Volume of a cone is one-third that of the corresponding cylinder.
• To find the volume of a cone, calculate as if it were a cylinder, then divide the result by three.

Example Calculation

1. Formula for volume of a cone:
   • Volume = (Area of base) × (Height) / 3
2. Given Dimensions:
   • Radius (r) = 5 cm
   • Height (h) = 10 cm
3. Area of base (circle):
   • A = πr² = π(5)² = 25π
4. Volume Calculation:
   • Volume = (25π × 10) / 3
   • Volume results in approximately 261.8 cm³ (assuming volume is in cubic centimeters).

Important Notes

• Always use the central height of the cone for volume calculations.
• Volume is always expressed in cubic units (e.g., cm³).