Volume of a Sphere Calculator

Published on: September 1, 2024
Final Answer: Free • Full Steps: Plus

This Volume of a Sphere Calculator finds the space inside a sphere from its radius, and shows the working line by line. A sphere fills two thirds of the cylinder that just contains it, which works out as the radius cubed and then multiplied by four thirds and by \(\pi\). The answer is written in cubic units and is usually rounded. Following the same pattern every time is what makes the volume of a sphere easy to check and repeat.

Sphere with radius labeled r

In the diagram, the radius is labeled \(r\).

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Cube the radius.
  5. Multiply by four thirds and by pi to get the volume.

Formula:

\(V = \frac{4}{3}\pi r^{3}\)
See Example 1 Hide Example 1

Problem

\(r = 7\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Cube the radius.
  5. Multiply by four thirds and by pi to get the volume.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the volume of a sphere whose radius is \(r = 7\).

\(r = 7\)

Step 2

Step 2 - Write the formula.

In this problem: A sphere fills two thirds of the cylinder that just contains it, which works out as the radius cubed multiplied by four thirds and by \(\pi\).

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

Step 3

Step 3 - Substitute the radius into the formula.

In this problem: Replacing \(r\) with \(7\) in \(V = \frac{4}{3}\pi r^{3}\) gives \(V = \frac{4}{3} \times \pi \times 7^{3}\).

\(V = \frac{4}{3} \times \pi \times 7^{3}\)

Step 4

Step 4 - Cube the radius.

In this problem: Cubing the radius gives \(7^{3} = 7 \times 7 \times 7 = 343\).

\(V = \frac{4}{3} \times \pi \times 343\)

Step 5

Step 5 - Multiply by four thirds and by pi to get the volume.

In this problem: Multiplying that by four thirds and by \(\pi\) gives \(\frac{4}{3} \times \pi \times 343 \approx 1436.76\).

\(V = \frac{4}{3} \times \pi \times 343 \approx 1436.76\)

Final Answer

\(V \approx 1436.76\)
See Example 2 Hide Example 2

Problem

\(r = 1.5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Cube the radius.
  5. Multiply by four thirds and by pi to get the volume.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the volume of a sphere whose radius is \(r = 1.5\).

\(r = 1.5\)

Step 2

Step 2 - Write the formula.

In this problem: A sphere fills two thirds of the cylinder that just contains it, which works out as the radius cubed multiplied by four thirds and by \(\pi\).

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

Step 3

Step 3 - Substitute the radius into the formula.

In this problem: Replacing \(r\) with \(1.5\) in \(V = \frac{4}{3}\pi r^{3}\) gives \(V = \frac{4}{3} \times \pi \times 1.5^{3}\).

\(V = \frac{4}{3} \times \pi \times 1.5^{3}\)

Step 4

Step 4 - Cube the radius.

In this problem: Cubing the radius gives \(1.5^{3} = 1.5 \times 1.5 \times 1.5 = 3.375\).

\(V = \frac{4}{3} \times \pi \times 3.375\)

Step 5

Step 5 - Multiply by four thirds and by pi to get the volume.

In this problem: Multiplying that by four thirds and by \(\pi\) gives \(\frac{4}{3} \times \pi \times 3.375 \approx 14.14\).

\(V = \frac{4}{3} \times \pi \times 3.375 \approx 14.14\)

Final Answer

\(V \approx 14.14\)

Example Problems

Click one to load it into the box and solve it.