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:

\( V = \dfrac{1}{3} \pi r^2 h \)

Using Diameter and Height:

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

The volume \( V \) can be calculated as:

\( V = \dfrac{1}{12} \pi d^2 h \)

Using Slant Height and Radius:

If the slant height \( s \) and the radius \( r \) are known,

The volume \( V \) can be calculated as:

\( V = \dfrac{1}{3} \pi r^2 \sqrt{s^2 – r^2} \)

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 \( h = \sqrt{s^2 – r^2} \)
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula

1. Formula for Volume Using Radius and Height

The formula \( V = \dfrac{1}{3} \pi r^2 h \) calculates the volume by finding the area of the circular base \( \pi r^2 \) 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 \( r = 4 \, \text{cm} \) and a height of \( h = 9 \, \text{cm} \).

Solution:

  1. Write down the formula: \( V = \dfrac{1}{3} \pi r^2 h \).
  2. Substitute \( r = 4 \) and \( h = 9 \): \( V = \dfrac{1}{3} \pi \times 4^2 \times 9 \).
  3. Calculate \( r^2 \): \( V = \dfrac{1}{3} \pi \times 16 \times 9 \).
  4. Multiply: \( V = \dfrac{144\pi}{3} = 48\pi \approx 150.8 \, \text{cm}^3 \) (using \( \pi \approx 3.14159 \)).

The volume of the cone is approximately \( 150.8 \, \text{cm}^3 \).

2. Formula for Volume Using Diameter and Height

The formula \( V = \dfrac{1}{12} \pi d^2 h \) 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 \( d = 10 \, \text{cm} \) and a height of \( h = 12 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{1}{12} \pi d^2 h \).
  2. Substitute \( d = 10 \) and \( h = 12 \): \( V = \dfrac{1}{12} \pi \times 10^2 \times 12 \).
  3. Calculate \( d^2 \): \( V = \dfrac{1}{12} \pi \times 100 \times 12 \).
  4. Multiply: \( V = \dfrac{1200\pi}{12} = 100\pi \approx 314.16 \, \text{cm}^3 \).

The volume of the cone is approximately \( 314.16 \, \text{cm}^3 \).

3. Formula for Volume Using Slant Height and Radius

The formula \( V = \dfrac{1}{3} \pi r^2 \sqrt{s^2 – r^2} \) calculates the volume by using the slant height \( s \). In this case, the vertical height \( h \) is calculated as \( h = \sqrt{s^2 – r^2} \), 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 \( r = 6 \, \text{cm} \) and a slant height of \( s = 10 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{1}{3} \pi r^2 \sqrt{s^2 – r^2} \).
  2. Substitute \( r = 6 \) and \( s = 10 \): \( V = \dfrac{1}{3} \pi \times 6^2 \times \sqrt{10^2 – 6^2} \).
  3. Calculate \( r^2 \) and \( s^2 – r^2 \): \( V = \dfrac{1}{3} \pi \times 36 \times \sqrt{100 – 36} \).
  4. Simplify inside the square root: \( V = \dfrac{1}{3} \pi \times 36 \times \sqrt{64} \).
  5. Find \( \sqrt{64} = 8 \): \( V = \dfrac{1}{3} \pi \times 36 \times 8 = 96\pi \approx 301.59 \, \text{cm}^3 \).

The volume of the cone is approximately \( 301.59 \, \text{cm}^3 \).

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.