Volume of a Cone Formula

The volume of a cone is the amount of space it occupies, which depends on the size of its circular base and its height. There are various ways to calculate the volume of a cone depending on the known measurements, such as radius and height, diameter and height, or slant height and radius. Below are all the formulas for calculating the volume of a cone, with detailed explanations and examples for each method.

Formulas for the Volume of a Cone

Using Radius and Height:

If the radius r of the base and the height h of the cone are known,

The volume V can be calculated with the formula:

V=13πr2h

Using Diameter and Height:

If the diameter d of the base and the height h are known,

The volume V can be calculated as:

V=112πd2h

Using Slant Height and Radius:

If the slant height s and the radius r are known,

The volume V can be calculated as:

V=13πr2s2r2

In these formulas:

  • r is the radius of the base of the cone
  • d is the diameter of the base of the cone, where d=2r
  • s is the slant height of the cone
  • h is the vertical height of the cone, where h=s2r2
  • π (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula


1 Formula for Volume Using Radius and Height

The formula V=13πr2h calculates the volume by finding the area of the circular base πr2 and multiplying it by the height h, then dividing by 3. This method is commonly used when both the radius and height of the cone are known.

Example 1: Calculating Volume with Radius and Height

Problem: Find the volume of a cone with a radius of r=4cm and a height of h=9cm.

Solution:

  1. Write down the formula: V=13πr2h.
  2. Substitute r=4 and h=9: V=13π×42×9.
  3. Calculate r2: V=13π×16×9.
  4. Multiply: V=144π3=48π150.8cm3 (using π3.14159).

The volume of the cone is approximately 150.8cm3.


2 Formula for Volume Using Diameter and Height

The formula V=112πd2h calculates the volume by using the diameter of the base instead of the radius. This formula is convenient if only the diameter is known, as it eliminates the need to convert the diameter to radius.

Example 2: Calculating Volume with Diameter and Height

Problem: A cone has a diameter of d=10cm and a height of h=12cm. Find its volume.

Solution:

  1. Write down the formula: V=112πd2h.
  2. Substitute d=10 and h=12: V=112π×102×12.
  3. Calculate d2: V=112π×100×12.
  4. Multiply: V=1200π12=100π314.16cm3.

The volume of the cone is approximately 314.16cm3.


3 Formula for Volume Using Slant Height and Radius

The formula V=13πr2s2r2 calculates the volume by using the slant height s. In this case, the vertical height h is calculated as h=s2r2, which allows us to find the volume using the radius and slant height directly.

Example 3: Calculating Volume with Slant Height and Radius

Problem: A cone has a radius of r=6cm and a slant height of s=10cm. Find its volume.

Solution:

  1. Write down the formula: V=13πr2s2r2.
  2. Substitute r=6 and s=10: V=13π×62×10262.
  3. Calculate r2 and s2r2: V=13π×36×10036.
  4. Simplify inside the square root: V=13π×36×64.
  5. Find 64=8: V=13π×36×8=96π301.59cm3.

The volume of the cone is approximately 301.59cm3.

These examples demonstrate how to calculate the volume of a cone using different known measurements, whether it’s the radius and height, diameter and height, or slant height and radius.