Volume Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Volume.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The amount of three-dimensional space that an object occupies, measured in cubic units such as cm³.

How many cubic centimetre blocks would it take to completely fill the inside of the object?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Volume counts how many unit cubes completely fill the inside of a 3D object, measured in cubic units.

Common stuck point: The procedure for volume is the easy part; the trap is stopping at length × width. Asking "Am I counting how many unit cubes fill a 3D solid?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting how many unit cubes fill a 3D solid?

Worked Examples

Example 1

easy

Find the volume of a rectangular prism with length

5

cm, width

3

cm, and height

4

cm.

Find the volume of this rectangular prism.

Answer

V = 60 \text{ cm}^3

First step

A rectangular prism (cuboid) has three mutually perpendicular dimensions. Its volume is the product of all three:

V = l \times w \times h

Full solution

2
Substitute the given dimensions — length $l = 5$ cm, width $w = 3$ cm, height $h = 4$ cm.
3
Compute: $V = 5 \times 3 \times 4 = 60$ cm³. Units are cubic (cm³) because length × length × length = length³.

Volume measures three-dimensional space. For a rectangular prism, it equals the product of its three dimensions. Volume is always in cubic units.

Example 2

medium

Find the volume of a cylinder with radius

3

cm and height

10

cm. Leave your answer in terms of

\pi

Find the volume in terms of π.

Example 3

medium

A cylindrical water tank has radius

3

m and height

5

m. Find its volume in cubic meters (leave answer in terms of

\pi

Example 4

medium

A cylinder has volume

100\pi

^3

and radius

5

cm. Find its height.

Volume = 100π cm³. Find h.

Example 5

medium

A triangular prism has a triangular base of area

12

^2

and length

8

cm. Find its volume.

Example 6

medium

A cube has volume

216

^3

. Find the side length.

Example 7

hard

A cone has volume

48\pi

^3

and height

9

cm. Find its radius.

Volume = 48π cm³. Find r.

Example 8

hard

A solid is formed by a cylinder (radius

2

cm, height

5

cm) with a hemisphere of radius

2

cm on top. Find the total volume in terms of

\pi

Example 9

hard

Water flows into a cylindrical tank (radius

2

m) at

4

^3

/min. How fast is the water level rising?

Example 10

hard

A rectangular box has volume

480

^3

. Its length is twice its width, and its height is

5

cm. Find the width.

Example 11

challenge

An ice cream cone has radius

3

cm and height

12

cm, topped with a hemisphere of ice cream of radius

3

cm. The ice cream melts and fills the cone completely. Does it overflow?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard

A rectangular storage box is

8

cm long and

6

cm wide. If its volume is

336

cm³, find its height.

Volume = 336 cm³. Find h.

Example 2

medium

A cylinder has volume

200\pi

cm³ and height

8

cm. Find its radius.

Volume = 200π cm³. Find r.

Example 3

easy

Find the volume of a box (rectangular prism) with dimensions

2 \times 3 \times 4

Find the volume of this box.

Example 4

easy

Find the volume of a cube with side

5

Example 5

easy

A box has volume

60

and base area

12

. Find its height.

Example 6

easy

Find the volume of a cylinder with radius

2

and height

5

. (In terms of

\pi

Find the volume in terms of π.

Example 7

easy

How many cubic centimeters in a

10 \times 10 \times 10

cm cube?

Example 8

easy

Find the volume of a box

1 \times 1 \times 1

Example 9

easy

If a cube's side doubles, what happens to its volume?

Example 10

easy

A fish tank is

40

cm long,

20

cm wide,

30

cm tall. What is its volume in cm³?

Fish tank — find the volume in cm³.

Example 11

medium

Find the volume of a triangular prism: triangular base with area

6

, length

10

Example 12

medium

Find the volume of a cylinder with diameter

6

and height

10

. (In terms of

\pi

Example 13

medium

Find the volume of a sphere with radius

3

. (In terms of

\pi

Find the volume in terms of π.

Example 14

medium

Find the volume of a cone with radius

4

and height

9

. (In terms of

\pi

Find the volume in terms of π.

Example 15

medium

A cylindrical water tank has radius

2

m and height

3

m. How many liters does it hold? (Use

\pi \approx 3.14

;

1

m³

= 1000

L.)

Example 16

medium

A cube has volume

27

. What is its surface area?

Example 17

medium

Find the volume of a square pyramid with base side

6

and height

10

Example 18

medium

Water fills a

5 \times 4 \times 10

tank to a depth of

6

. What volume of water?

Example 19

medium

A cylinder and cone have the same radius and height. The cylinder's volume is

30\pi

. What's the cone's volume?

Example 20

challenge

A sphere of radius

r

fits exactly inside a cube. The cube has volume

64

. Find the sphere's volume in terms of

\pi

Example 21

challenge

A rectangular box has volume

48

, and its dimensions are three consecutive even integers. Find them.

Example 22

challenge

Two cubes have volumes

8

and

27

. What is the ratio of their surface areas?

Example 23

easy

A rectangular prism is

7

cm long,

5

cm wide, and

2

cm tall. Find its volume.

Find the volume of this rectangular prism.

Example 24

easy

A cube has side

4

cm. Find its volume.

Example 25

easy

Find the volume of a cylinder with radius

1

cm and height

10

cm. Leave in terms of

\pi

Example 26

easy

A storage box is

10

\times 10

\times 5

cm. Find its volume.

Example 27

easy

A juice carton is

4

\times 4

\times 8

in. What is its volume in cubic inches?

Example 28

medium

Find the volume of a cone with radius

3

cm and height

7

cm. Leave in terms of

\pi

Find the volume in terms of π.

Example 29

medium

Find the volume of a sphere with radius

3

cm. Leave in terms of

\pi

Find the volume in terms of π.

Example 30

medium

A box has volume

144

^3

, length

6

cm, and width

4

cm. Find its height.

Volume = 144 cm³. Find h.

Example 31

medium

A swimming pool is

25

m long,

10

m wide, and

2

m deep. How many cubic meters of water does it hold?

Example 32

medium

A rectangular fish tank is

40

\times 30

\times 25

cm. Find its volume in liters (1 L = 1000 cm

^3

Example 33

hard

A sphere has volume

\frac{32}{3}\pi

^3

. Find its radius.

Example 34

hard

A rectangular prism of dimensions

6 \times 4 \times 3

has a cylindrical hole of radius

1

drilled through its longest dimension. Find the remaining volume in terms of

\pi

Example 35

hard

A pyramid has a square base of side

6

cm and height

10

cm. Find its volume.

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Related Concepts

Area Multiplication Surface Area

Background Knowledge

These ideas may be useful before you work through the harder examples.

areamultiplication