Volume of a Sphere Formula

The volume of a sphere is the amount of space it occupies, determined by its radius, diameter, or circumference.

Below are all the formulas for calculating the volume of a sphere, with detailed explanations and examples for each method.

Formulas for the Volume of a Sphere

Using Radius:

If the radius \( r \) of the sphere is known,

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

\( V = \dfrac{4}{3} \pi r^3 \)

Using Diameter:

If the diameter \( d \) of the sphere is known,

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

\( V = \dfrac{1}{6} \pi d^3 \)

Using Circumference:

If the circumference \( C \) of the sphere is known,

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

\( V = \dfrac{C^3}{6\pi^2} \)

In these formulas:

  • \( r \) is the radius of the sphere
  • \( d \) is the diameter of the sphere, where \( d = 2r \)
  • \( C \) is the circumference of the sphere, where \( C = 2\pi r \)
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula

1. Formula for Volume Using Radius

The formula \( V = \dfrac{4}{3} \pi r^3 \) calculates the volume by cubing the radius, multiplying by \( \pi \), and then by \( \dfrac{4}{3} \). This method is commonly used when the radius of the sphere is known.

Example 1: Calculating Volume with Radius

Problem: Find the volume of a sphere with a radius of \( r = 6 \, \text{cm} \).

Solution:

  1. Write down the formula: \( V = \dfrac{4}{3} \pi r^3 \).
  2. Substitute \( r = 6 \): \( V = \dfrac{4}{3} \pi \times 6^3 \).
  3. Calculate \( r^3 \): \( V = \dfrac{4}{3} \pi \times 216 \).
  4. Multiply: \( V = 288\pi \approx 904.32 \, \text{cm}^3 \) (using \( \pi \approx 3.14159 \)).

The volume of the sphere is approximately \( 904.32 \, \text{cm}^3 \).

2. Formula for Volume Using Diameter

The formula \( V = \dfrac{1}{6} \pi d^3 \) calculates the volume using the diameter instead of the radius. This is convenient if only the diameter is known, as it avoids the need to divide by 2 to find the radius.

Example 2: Calculating Volume with Diameter

Problem: A sphere has a diameter of \( d = 10 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{1}{6} \pi d^3 \).
  2. Substitute \( d = 10 \): \( V = \dfrac{1}{6} \pi \times 10^3 \).
  3. Calculate \( d^3 \): \( V = \dfrac{1}{6} \pi \times 1000 \).
  4. Multiply: \( V = \dfrac{1000\pi}{6} \approx 523.6 \, \text{cm}^3 \).

The volume of the sphere is approximately \( 523.6 \, \text{cm}^3 \).

3. Formula for Volume Using Circumference

The formula \( V = \dfrac{C^3}{6\pi^2} \) calculates the volume using the circumference of the sphere. This formula is useful if only the circumference is known, as it eliminates the need to find the radius first.

Example 3: Calculating Volume with Circumference

Problem: A sphere has a circumference of \( C = 15 \, \text{cm} \). Find its volume.

Solution:

  1. Write down the formula: \( V = \dfrac{C^3}{6\pi^2} \).
  2. Substitute \( C = 15 \): \( V = \dfrac{15^3}{6\pi^2} \).
  3. Calculate \( C^3 \): \( V = \dfrac{3375}{6\pi^2} \).
  4. Approximate \( \pi^2 \approx 9.8696 \): \( V = \dfrac{3375}{6 \times 9.8696} \approx 57.13 \, \text{cm}^3 \).

The volume of the sphere is approximately \( 57.13 \, \text{cm}^3 \).

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