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# Volume of a Sphere

You may recall that the volume of a three-dimensional object is the amount of space it occupies. You may also recall that a sphere is a set of points in space that are a given distance r from its center. We can use both of these definitions to calculate the volume of a sphere!

The formula for the volume of a sphere is $V=\frac{4}{3}\pi {r}^{3}$ where r is the radius of the sphere. Here is an illustration of what a sphere might look like in a math problem:

Generally, you will be able to sub in 3.14 as the value of π. Alternatively, you may be asked to leave your answer in terms of pi (meaning the π symbol will appear in your answer).

Volume is measured in cubic units, like ${\mathrm{in}}^{3}$ , ${m}^{3}$ , ${\mathrm{ft}}^{3}$ , ${\mathrm{cm}}^{3}$ , and so on. Some volume questions might try to trick you by giving you measurements in different units, so pay close attention to what you're doing! Likewise, some problems will give you the diameter of the sphere instead of the radius. Fortunately, you already know that the radius is half of the diameter.

## Calculating the volume of a sphere

The best way to study the volume of a sphere is to do a few practice problems, so let's look at an example. Here is our sphere:

We know that the formula for the volume of a sphere is $\frac{4}{3}\pi {r}^{3}$ . The diagram above tells us that r is 8, and we can approximate the π symbol with 3.14. That gives us:

$\frac{4}{3}×3.14×{8}^{3}$

$\frac{4}{3}×3.14×512$

After a little bit of multiplication, we have an answer of approximately $2144\phantom{\rule{4pt}{0ex}}{m}^{3}$ . That wasn't so bad, was it?

## Fun facts about the volume of a sphere

If you know the volume of a sphere, you can easily figure out a few other things. For instance, the volume of a hemisphere is one-half the volume of the associated sphere. This can be expressed mathematically as $\frac{1}{2}V$ . Furthermore, the volume of a sphere is $\frac{2}{3}$ the volume of a cylinder with the same radius and height equal to the diameter.

## Practice problems on the volume of a sphere

a. What is the volume of a sphere with a radius of 3 inches? Use 3.14 for pi and round your answer to the nearest square inch.

The formula for the volume of a sphere is $V=\frac{4}{3}\pi {r}^{3}$ . Given r is 3 inches, and using 3.14 for pi:

$V=\frac{4}{3}×3.14×{3}^{3}$

$V=\frac{4}{3}×3.14×27$

$V\approx 113.04$

Rounded to the nearest sq. in, the volume is about $113{\mathrm{in}}^{3}$ .

b. What is the volume of the sphere with a radius of 5 meters? Express your answer in terms of pi and round to the nearest sq. meter.

Given r is 5 meters, using the formula for the volume of a sphere:

$V=\frac{4}{3}\pi ×{5}^{3}$

$V=\frac{4}{3}\pi ×125$

$V\approx 166.67\pi$

c. What is the volume of a sphere with a diameter of 20 feet? Use 3.14 for pi and round your answer to the nearest sq. ft.

First, find the radius by dividing the diameter by 2: $r=\frac{20}{2}=10$ feet. Then, using the formula for the volume of a sphere:

$V=\frac{4}{3}×3.14×{10}^{3}$

$V=\frac{4}{3}×3.14×1000$

$V\approx 4186.8$

Rounded to the nearest integer, the volume is about $4187{\mathrm{ft}}^{3}$ .

Sphere

## Varsity Tutors helps students with the volume of a sphere

Finding the volume of a sphere can be challenging, but geometry only becomes more difficult as students advance. If the student in your life is struggling to remember the formula for the volume of a sphere or how to apply it correctly, a 1-on-1 math tutor could identify why they are struggling and address the issue at its root cause. Please reach out to the friendly Educational Directors at Varsity Tutors for further information.

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