# Volume of a Sphere

## Volume of a Sphere – Introduction

Have you ever wondered, “I can draw a circle, but I cannot draw a sphere? Why?” This is because a circle is a two-dimensional figure and does not have volume, whereas a sphere is a three-dimensional shape with no edges or vertices. That means its points lie in space. Hence, you cannot draw it. This is the reason we always find the volume of a sphere to calculate the amount of space it occupies.

Scroll ahead to learn about the volume of a sphere formula, the derivation of the sphere volume formula, some solved examples, facts, and more.

## What Is the Volume of a Sphere?

Wondering how we can find the volume of a sphere? Hold on, we will get to that, but first, understand what the volume of a sphere means. The volume of a sphere is the measure of three-dimensional space occupied by a sphere. It depends on the sphere’s radius, which is half the diameter (the longest line inside the sphere that passes through the center of the sphere).

That means if the radius of the sphere changes, its volume changes too!

The volume of a sphere is measured in cubic units, such as $m^3$, $cm^3$, and so on.

## What Is the Formula to Find the Volume of a Sphere?

How do you find the volume of a sphere given that “r” is the radius of the sphere? The volume of a sphere equation is as follows:

The volume of a sphere $= \frac{4}{3} \pi r^3$

## How to Calculate the Volume of a Sphere?

Suppose the radius of the sphere is 6 cm, then its volume will be:

As we know, the volume of the sphere, V $= \frac{4}{3} \pi r^3$

Here, $r = 6$ cm

Thus, volume of sphere, V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 6 \times 6 \times 6$

$\Rightarrow$ V $= 904.32 cm^3$

Therefore, the volume of a sphere is 904.32 cm3

## Volume of Solid Sphere

If the radius of the solid sphere is r, the volume of the sphere is given by:

Volume of Sphere, V $= \frac{4}{3} \pi r^3$

## Volume of Hollow Sphere

Sphere that has a cavity or space inside is called a hollow sphere.

Let the radius of the outer sphere is R and the radius of the inner sphere is r.

The volume of the sphere, V is given by:

Volume of Sphere, V = Volume of Outer Sphere – Volume of Inner Sphere

$= \frac{4}{3} \pi R^3 – \frac{4}{3} r^3$

$= \frac{4}{3} \pi (R^3 – r^3)$

How about some fun and interesting facts about the sphere? Let’s take a look!

1. A sphere is symmetrical and round. It does not have any faces, corners, or edges.
2. Balls, marbles, and even Earth are shaped like spheres.
3. A hemisphere is an exact half of a sphere.
4. All the sphere’s surface points are the same distance “r” from the center.
5. The sphere appears in nature when a surface wants to be as small as possible. For example, if you blow up a balloon, it naturally forms a sphere!

## Conclusion

So, how was the lesson? We hope you have got a clear understanding of the volume of spheres. Stay tuned with SplashLearn for more useful math lessons like these.

## Solved Examples

1. What is the volume of a sphere with a radius of 12 units?

Solution: To solve for the volume of a sphere, you must first know the equation for the volume of a sphere.

The equation for the volume of a sphere is:

V $= \frac{4}{3} \pi r^3$

If the radius of the sphere is 12, then you plug that in for r and solve:

V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 12 \times 12 \times 12 = 7234.56$ cubic units

2. Find the volume of the sphere whose diameter is 28 cm.

Solution: Given, diameter $= 28 cm$

So, radius $= \frac{Diameter}{2} = \frac{28}{2} = 14 cm$

From there, we can use the formula, which is

V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 14 \times 14 \times 14 = 11488.23 cm^3$

3. Find the volume of a spherical tank whose radius is 3 inches.

Solution:

The equation for the volume of a sphere is:

V $= \frac{4}{3} \pi r^3$

If the radius of the sphere is 3 inches, then you plug that in for r and solve:

V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 3 \times 3 \times 3 = 113.04$ cubic inches

4. Nina created a spherical ball of radius 5 cm using clay. She then cut the sphere into two equal parts. What will be the volume of each part?

Solution:

The equation for the volume of a sphere is:

V $= \frac{4}{3} \pi r^3$

If the radius of the sphere is 5 cm, then you plug that in for r and solve:

V $= \frac{4}{3} \pi r^3 = 43 3.14 5 5 5 = 523.33$ $cm^3$

Hence volume of each cut part $= \frac{523.33}{2} = 261.66$ $cm^3$

5.  Sam wants to create a sphere of radius 7 cm using some clay. What will be the volume of the sphere formed?

Solution:

The equation for the volume of a sphere is:

V $= \frac{4}{3} \pi r^3$

If the radius of the sphere is 7 cm, then you plug that in for r and solve:

V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 7 \times 7 \times 7 = 1436.02$ $cm^3$

## Practice Problems

1

### What is the volume of the sphere, whose radius is 4 units?

$88.3\pi$
$77.3\pi$
$85.3\pi$
$90\pi$
CorrectIncorrect
Correct answer is: $85.3\pi$
To solve for the volume of a sphere, you must first know the equation for the volume of a sphere. The equation is V $= \frac{4}{3} \pi r^3$ Then plug the radius into the equation for r, yielding V $= \frac{4}{3} \times \pi \times 4 \times 4 \times 4 = 85.3\pi$
2

### Find the volume of a sphere of radius 10 cm.

$4186.66 cm^3$
$2186.66 cm^3$
$4189.66 cm^3$
$41.66 cm^3$
CorrectIncorrect
Correct answer is: $4186.66 cm^3$
The volume of the sphere $=$ V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 10 \times 10 \times 10 = 4186.66 cm^3$
3

### A spherical-shaped tank has a radius of 21 m. Now find the capacity of it in liters to store water in it.

28,808,000 liters
38,808,000 liters
38,888,000 liters
30,808,000 liters
CorrectIncorrect
The given values are,
r $= 21$ m
Now the volume of a sphere,
V $= \frac{4}{3} \times \pi \times r^3$
V $= \frac{4}{3} \times 21 \times 21 \times 21$
V $= 4 \times 22 \times 21 \times 21$
V $= 38808$ $m^3$
Since,
$1 m^3 = 1000$ liter
Then the capacity of the tank,
$= 38,808 m^3 \times 1000$
$= 38,808,000$ liter
So, 38,808,000 liters of water can be stored in the tank.
4

### The volume of the sphere is $2100 cm^3$. What is the radius of the hemisphere formed by cutting the sphere into two equal parts?

$1200 cm^3$
$1100 cm^3$
$1050 cm^3$
$2100 cm^3$
CorrectIncorrect
Correct answer is: $1050 cm^3$
Volume of hemisphere $= \frac{Volume of sphere}{2} = 1050 cm^3$
5

### What is the volume of a sphere with a radius of 18 cm?

$8004.78 cm^3$
$9304.88 cm^3$
$3908.78 cm^3$
$3052.08 cm^3$
CorrectIncorrect
Correct answer is: $3052.08 cm^3$
Diameter of the sphere $= 18 cm$
Radius of the sphere $= \frac{18}{2} = 9 cm$
The volume of the sphere $=$ V $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 9 \times 9 \times 9 = 3052.08 cm^3$

The general formula for the volume of a sphere is given as V $= \frac{4}{3} \pi r^3$ . Let’s say “d” is its diameter. According to the definition of diameter, we have d $= 2r$. From this, we get the value of radius $= \frac{d}{2}$. Substituting this in the formula, the volume of a sphere can be found.
For a sphere, the surface area is S $=$ 4*π*R*R, where R is the sphere’s radius, and π is 3.1415… The volume of a sphere is V $= \frac{4 \times R \times R \times R}{3}$. So for a sphere, the surface area to volume ratio is given by: $\frac{S}{V}=\frac{3}{R}$.
The volume of a sphere is measured in cubic units, such as $m^3, cm^3,$ etc.