Volume of cones

key notes :

Formula for Volume of a Cone:

• The volume V of a cone is given by the formula:

V = 1/3 πr²h

where:

• r is the radius of the base of the cone,
• h is the height of the cone,
• π\piπ is approximately 3.1416.

Understanding the Formula:

• The volume of a cone is one-third of the volume of a cylinder with the same radius and height.
• This is because the cone tapers to a point at the top, reducing the amount of space it occupies compared to a cylinder.

Key Components:

• Radius (r): The distance from the center of the base to its edge.
• Height (h): The perpendicular distance from the apex (top point) to the base.

Units:

• The volume will be in cubic units, depending on the units used for the radius and height. For example, if r and h are in centimeters, the volume will be in cubic centimeters (cm³).

Example:

• If a cone has a radius of 5 cm and a height of 12 cm, the volume is:

V = 13π (5²) (12) = 1/3π (25)(12) = 100π ≈ 314.16 cm³

Surface Area Relation:

• The volume is related to the surface area but is calculated separately. The surface area includes both the base and the slanted surface area of the cone.

Real-life Example Problems:

• Find the volume of a cone when given its height and radius.
• Compare the volumes of cones with different dimensions to understand how changes in height or radius affect the overall volume.

Volume and Similar Cones:

• If two cones are similar, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions (e.g., the ratio of their heights or radii).

Learn with an example

let’s practice!