Volume of a Cylinder Formula

The volume of a cylinder is the amount of space it occupies, calculated based on its circular base and height. There are several ways to determine the volume depending on the measurements available, such as radius and height, diameter and height, circumference and height, or circular area and height. Below are all the possible formulas for calculating the volume of a cylinder, with detailed explanations and examples for each method.

Formulas for the Volume of a Cylinder

Using Radius and Height:

If the radius \( r \) of the base and the height \( h \) of the cylinder are known,

The volume \( V \) can be calculated with the formula:

\( V = \pi r^2 h \)

Using Diameter and Height:

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

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

\( V = \dfrac{\pi d^2 h}{4} \)

Using Circumference and Height:

If the circumference \( C \) of the base and the height \( h \) are known,

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

\( V = \dfrac{C^2 h}{4\pi} \)

Using Circular Area and Height:

If the area of the circular base \( A \) and the height \( h \) are known,

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

\( V = A \times h \)

In these formulas:

  • \( r \) is the radius of the base of the cylinder
  • \( d \) is the diameter of the base of the cylinder, where \( d = 2r \)
  • \( C \) is the circumference of the base, where \( C = 2\pi r \)
  • \( A \) is the area of the circular base, where \( A = \pi r^2 \)
  • \( h \) is the height of the cylinder
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula

1. Formula for Volume Using Radius and Height

The formula \( V = \pi r^2 h \) calculates the volume of a cylinder by finding the area of the circular base, \( \pi r^2 \), and multiplying it by the height \( h \). This method is commonly used when both the radius and height of the cylinder are known.

Example 1: Calculating Volume with Radius and Height

Problem: Find the volume of a cylinder with a radius of \( r = 4 \, \text{cm} \) and a height of \( h = 10 \, \text{cm} \).

Solution:

  1. Write down the formula: \( V = \pi r^2 h \).
  2. Substitute \( r = 4 \) and \( h = 10 \): \( V = \pi \times 4^2 \times 10 \).
  3. Calculate \( r^2 \): \( V = \pi \times 16 \times 10 \).
  4. Multiply: \( V = 160\pi \approx 502.65 \, \text{cm}^3 \) (using \( \pi \approx 3.14159 \)).

The volume of the cylinder is approximately \( 502.65 \, \text{cm}^3 \).

2. Formula for Volume Using Diameter and Height

The formula \( V = \dfrac{\pi d^2 h}{4} \) calculates the volume by using the diameter of the base instead of the radius. This is convenient if only the diameter and height are known.

Example 2: Calculating Volume with Diameter and Height

Problem: A cylinder has a diameter of \( d = 8 \, \text{cm} \) and a height of \( h = 12 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{\pi d^2 h}{4} \).
  2. Substitute \( d = 8 \) and \( h = 12 \): \( V = \dfrac{\pi \times 8^2 \times 12}{4} \).
  3. Calculate \( d^2 \): \( V = \dfrac{\pi \times 64 \times 12}{4} \).
  4. Multiply and divide: \( V = \dfrac{768\pi}{4} = 192\pi \approx 603.19 \, \text{cm}^3 \).

The volume of the cylinder is approximately \( 603.19 \, \text{cm}^3 \).

3. Formula for Volume Using Circumference and Height

The formula \( V = \dfrac{C^2 h}{4\pi} \) calculates the volume using the circumference of the circular base. This method is useful if the circumference and height are known, and it is derived by expressing the radius in terms of the circumference \( C = 2\pi r \).

Example 3: Calculating Volume with Circumference and Height

Problem: A cylinder has a circumference of \( C = 15 \, \text{cm} \) and a height of \( h = 20 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{C^2 h}{4\pi} \).
  2. Substitute \( C = 15 \) and \( h = 20 \): \( V = \dfrac{15^2 \times 20}{4\pi} \).
  3. Calculate \( C^2 \): \( V = \dfrac{225 \times 20}{4\pi} = \dfrac{4500}{4\pi} \).
  4. Divide and approximate: \( V = \dfrac{4500}{12.566} \approx 358.14 \, \text{cm}^3 \).

The volume of the cylinder is approximately \( 358.14 \, \text{cm}^3 \).

4. Formula for Volume Using Circular Area and Height

The formula \( V = A \times h \) calculates the volume of a cylinder if the area \( A \) of the circular base and the height \( h \) are known. This formula is derived from the fact that the volume of a cylinder is the area of the base times its height.

Example 4: Calculating Volume with Circular Area and Height

Problem: A cylinder has a circular base area of \( A = 50 \, \text{cm}^2 \) and a height of \( h = 8 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = A \times h \).
  2. Substitute \( A = 50 \) and \( h = 8 \): \( V = 50 \times 8 \).
  3. Multiply: \( V = 400 \, \text{cm}^3 \).

The volume of the cylinder is \( 400 \, \text{cm}^3 \).

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