# Volume of a cube – Volume of a cube — Math Open Reference

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## Volume of a cube — Math Open Reference

Volume enclosed by a cube

Definition:
The number of cubic units that will exactly fill a cube

Try this
Drag the orange dot to resize the cube. The volume is calculated as you drag.

### How to find the volume of a cube

Recall that a cube has all edges the same length (See Cube definition).
The volume of a cube is found by multiplying the length of any edge by itself twice.
So if the length of an edge is 4, the volume is 4 x 4 x 4 = 64

Or as a formula:

 volume = s3 where:s  is the length of any edge of the cube.

In the figure above, drag the orange dot to resize the cube.
From the edge length shown, calculate the volume of the cube and verify that it agrees with the calculation in the figure.

When we write volume = s3, strictly speaking this should be read as «s to the power 3»,
but because it is used to calculate the volume of cubes it is usually spoken as «s cubed».

### Calculator

Use the calculator on the right to calculate the properties of a cube.

Enter any one value and the others will be calculated. For example, enter the side length and the volume will be calculated.

Similarly, if you enter the surface area, the side length needed to get that area will be calculated.

### Some notes on the volume of a cube

Recall that a cube is like an empty box. It has nothing inside, and the walls of the box have zero thickness.
So strictly speaking, the cube has zero volume.
When we talk about the volume of a cube, we really are talking about how much liquid it can hold, or
how many unit cubes would fit inside it.

Think of it this way: if you took a real, empty metal box and melted it down, you would end up with a small blob of metal.
If the box was made of metal with zero thickness, you would get no metal at all. That is what we mean when we say a cube has no volume.

The strictly correct way of saying it is «the volume enclosed by a cube» — the amount space there is inside it.
But many textbooks simply say «the volume of a cube» to mean the same thing.
However, this is not strictly correct in the mathematical sense.
What they usually mean when they say this is the volume enclosed by the cube.

### Units

Remember that the length of an edge and the volume will be in similar units.
So if the edge length is in miles, then the volume will be in cubic miles, and so on.

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## How do you find the Surface Area and Volume of a Cube

### How do you find the Surface Area and Volume of a Cube

If the length of each edge of a cube is ‘a’ units, then

1. Total surface area of the cube = 6a2  sq. units.
2. Lateral surface area = 4a2
3. Volume of the cube = a3 cubic units
4. Diagonal of the cube = a units.
5. Length of its diagonals = a√3
6. Total length of its edges = 12a

### Surface Area and Volume of a Cube Example Problems with Solutions

Example 1:    If each edge (side) of a cube is 8 cm ; find its surface area and lateral surface area.
Solution:     Given each side of the cube (a) = 8 cm
∴    Its surface area = 6a2 = 6 × 82 sq. cm
= 6 × 64 cm2  = 384 cm2
Lateral surface area = 4a2 = 4 × 82 sq. cm
= 4 × 64 cm2  = 256 cm2

Example 2:    A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
(i)   Which box has the greater lateral surface area and by how much ?
(ii)  Which box has the smaller total surface area and by how much ?
Solution:     (i)   For the cubical box :
Each edge = 10 cm i.e., a = 10 cm
∴    Lateral surface area of the cubical box
= 4a2 = 4 × 102 cm2 = 400 cm2
For the cuboidal box :
ℓ = 12.5 cm, b = 10 cm & h = 8 cm
∴    Lateral surface area of the cuboidal box
= 2(ℓ + b) × h
= 2(12.5 + 10) × 8 cm2
= 2 × 22.5 × 8 cm2 = 360 cm2
Clearly, cubical box has greater lateral surface area by
400 cm2 – 360 cm2  = 40 cm2
(ii)  Total surface area of the cubical box:
= 6 a2  = 6 × 102 sq. cm = 600 cm2
Total surface area of the cuboidal box
= 2(ℓ × b + b × h + h × ℓ)
= 2(12.5 × 10 +10 × 8 + 8 × 12.5) cm2           = 2(125 + 80 + 100) cm2 = 610 cm2
Clearly, cubical box has smaller surface area by
610 cm2 – 600 cm2 = 10 cm2

Example 3:    Find the volume of a solid cube of side 12 cm. If this cube is cut into 8 identical cubes, find :
(i)   Volume of each small cube.
(ii)  Side of each small cube.
(iii) Surface area of each small cube.
Solution:     Since, side (edge) of the given solid cube = 12 cm.
∴    Volume of given solid cube = (edge)3
= (12 cm)3 = 1728 cm3   Ans.
(i)   As the given cube is cut into 8 identical cubes.
⇒   Vol. of 8 small cubes obtained
= Vol. of given cube = 1728 cm3
⇒   Volume of each small cube
=  = 216 cm3
(ii)  If edge (side) of each small cube = x cm
(edge)3 = Volume
⇒   x3 = 216 = 6 × 6 × 6 = 63 ⇒   x = 6 cm
∴    Side of each small cube = 6 cm
(iii) Surface area of each small cube
= 6 × (edge)2
= 6 × (6 cm)2 = 216 cm2

Example 4:    A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute ?
Solution:     Volume of water that flows through a river, canal or pipe, etc., in unit time
=    Area of cross-section × Speed of water through it.
x km/hr = x ×
Reason : 1 km/hr =
Since, area of cross-section of the river
= Its depth × its width
= 3m × 40m = 120 m2
And, speed of flow of water through the river
= 2 km/hr = 2 ×
∴ Vol. of water that flows through it in 1 sec.
=  Area of cross-section × speed of water through it.
= 120 ×
⇒   Vol. of water that flows through it in
1 min. (60 sec.)
=  = 4000 m3
⇒   Vol. of water that will fall into the sea in a minute. = 4000 m3

Example 5:    The volume of a cube is numerically equal to its surface area. Find the length of its one side.
Solution:     Let length of each side is a unit.
Given: Volume of the cube = Surface area of the cube.
⇒   a3 = 6a2        ⇒   a = 6
∴    The length of one side of the cube = 6 cm

Example 6:    A solid cuboid has square base and height
12 cm. If its volume is 768 cm3, find :
(i)   side of its square base.
(ii)  surface area.
Solution:     (i)   Let side of the square base be x cm
i.e., ℓ = b = x cm
ℓ × b × h = volume
⇒   x × x × 12 = 768
[Given, height = 12 cm]
⇒   x2 =  ⇒   x = √64cm = 8 cm.
∴    Side of the square base = 8 cm
(ii)  Now, ℓ = 8 cm, b = 8 cm and h = 12 cm
∴  Surface area = 2(ℓ × b + b × h + h × ℓ)
= 2(8 × 8 + 8 × 12 + 12 × 8) cm2 = 512 cm2

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## Volume of a Cube — The Basics and Examples

00:00:04.170
In this lesson, we will learn how to find the volume of a cube.

00:00:09.000
First, consider this square with each side, with the length, L.

00:00:14.210
Now, we should already know that the area, A of this square is LxL.

00:00:21.110
Next, let’s change this square into a cube.

00:00:25.190
Note that, all the sides of a cube have the same length, L.

00:00:30.220
To find the volume of the cube, we just need to multiply the area, A with L.

00:00:37.200
Hence, by multiplying these three ‘L’s together, we get L cube.

00:00:43.010
Therefore, the formula for the volume is, V = L cube.

00:00:52.190
Now, it is important that we include the unit for volume.

00:00:56.210
Since the unit for L is not given, we can write the unit for this volume as cubic unit.

00:01:03.040
Alright, let’s take a look at some examples on how to find the volume of a cube.

00:01:08.220
Find the volume of this cube when the length of each side is 5cm.

00:01:14.060
To solve this, we need to use the formula, V = L cube.

00:01:19.120
Now, since we know the length of each side of this cube is 5cm, we can substitute L with 5.

00:01:28.230
Next, 5 to the power of 3 is equals to, 5 multiply by 5 multiply by 5. This gives 125. Let’s write it here.

00:01:41.090
Now, we have the volume as 125. Note that, this number is meaningless, unless we include the unit for it.

00:01:51.030
Since the unit for the length is in centimeter, the unit for the volume will be cubic centimeter.

00:01:57.070
Hence, the volume is 125 cubic cm.

00:02:04.010
Next example, if the volume of the cube is 8 cubic ft, find the length of each side.

00:02:10.180
Now, let’s start with the formula, V equals to L cube, where ‘L’ is the length of each side.

00:02:18.150
Since the volume of the cube is 8, we can substitute V with 8.

00:02:23.210
So, now we have, 8 = L cube.

00:02:28.000
To find ‘L’, we can see that since 8 is equals to ‘L’ cube, L can be found by calculating the cube root of 8.

00:02:38.040
Before we continue, let’s rewrite this equation properly by swapping these terms.

00:02:45.080
Next, cube root of 8 is 2. So we have, L equals to 2.

00:02:52.120
Now, this number is meaningless, unless we include the unit for it.

00:02:57.210
Since the volume is in cubic feet, the length of the sides will be in feet.

00:03:03.000
Hence, the length of the sides is L = 2 ft.

00:03:08.130
That is all for this lesson. Try out the practice question to test your understanding.

www.mathexpression.com

## Volume of Cube — Formulae

### Volume of Cube Formula

free Cube Volume Calculator

`V = S3`
• V is the volume enclosed by the cube
• S is the side length, it is also commonly represented by an A in other problem sets

This simple cube volume formula applies only to true cubes where all sides are an equal length. If all sides are not an equal length, but still parallel, please use the Length x Width x Height formula below.

### Volume of a Rectangular Prism

free Rectangular Prism Volume Calculator

`V = L * W * H`
• V is the volume enclosed by the rectangular prism
• L is the side length
• W is the side width
• H is the side height

For the purposes of the calculation it will not matter which sides are considered to be the Length, Width, or Height. However, all Length lines must run parallel to each other, as must all Width and Height lines to their respective partners.

### Surface Area of a Cube

free Cube Surface Area Calculator

`A = 6S2`
• A is the surface area of the rectangular prism
• S is the side length

A cube is comprised of six equal squares. To find the total surface area first find the surface Area of a square by multiplying Side length by Side length, this is the same as saying Side2. Then multiplying the area of one square (a face) by the total number of faces (six), we now know the total surface area of the cube.

### Face Diagonal Length

free Cube Face Diagonal Length Calculator

`L = √2S`
• L is the length of a diagonal line across a cube’s face (through a square)
• S is the side length

This formula calculates the length of a diagonal line running across a single cube face from a vertex to another vertex that forms the opposite corner, this is the longest line that can be draw across a cube’s face. Since the face of a cube is a square is formula also applies to diagonals across the face of a square and is essentially a two dimensional problem.

### Volume Diagonal Length

free Cube Volume Diagonal Length Calculator

`L = √3S`
• L is the maximum length of a diagonal through a cube
• S is the side length

This formula calculates the maximum length of a diagonal line running through cube volume from a vertex to another vertex that forms the opposite corner, this is the longest line that can be draw through a cube’s volume.

### Inscribed Sphere Radius

free Inscribed Sphere Radius Calculator

`R = S/2`
• R is the maximum radius of a sphere inscribed in a cube
• S is the side length of the cube

A sphere that is inscribed in a cube is one that mathematically touches the face of the cube without penetrating any part of the cube. In mathematics the term tangent is used to refer to two entities that touch in one place without crossing. Therefore, the largest sphere that can be inscribed in a cube is tangential to the faces of the cube (it doesn’t stick through the cube anywhere, it is totally inside the cube).

### Sphere Radius Tangent to Cube Edges

free Radius of a Sphere Tangent to Cube Edge Calculator

`R = S/√2`
• R is the radius of a sphere tangent to the edges of a cube
• S is the side length of the cube

A sphere that touches the edges of a cube only once on every side of the cube is said to be tangent to the edges of the cube. This sphere will poke through the faces of the cube but not be large enough to enclose the entire cube as the cube corners will poke out of the sphere.

### Circumscribed Sphere Radius

free Circumscribed Sphere Radius Calculator

`R = S * √3/2`
• R is the radius of a sphere that circumscribes a cube
• S is the side length of the cube

A sphere that completely contains a cube and touches each of its vertices is said to circumscribed the cube.

### Inscribed Sphere Volume

free Inscribed Sphere Radius Calculator

`V = (4/3)π(S/2)3`
• V is the volume of a sphere inscribed in a cube
• S is the side length of the cube
• π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

An inscribed sphere is completely contained by a cube touching only its faces. Previously we found the formula for the radius of an inscribed sphere, recall it was R = S/2. The volume of a sphere can be found using the formula V = (4/3)πR3. To complete the formula substitute the radius R with our method for finding the radius of an inscribed sphere.

### Sphere Volume Tangent to Cube Edges

free Volume of a Sphere Tangent to Cube Edge Calculator

`V = (4/3)π(S/√2)3`
• R is the radius of a sphere tangent to the edges of a cube
• S is the side length of the cube
• π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

This formula substitutes the formula for finding the radius of a sphere tangent to the edge of a cube for the radius term in the formula for finding the volume of a sphere.

### Circumscribed Sphere Volume

free Circumscribed Sphere Volume Calculator

`V = (4/3)π(S * √3/2)3`
• R is the radius of a sphere that circumscribes a cube
• S is the side length of the cube
• π (Pi) is a mathematical constant defined by the ratio of the circumference of a circle to the diameter of a circle

Substitute the formula for finding the radius of a sphere that circumscribes a cube for the radius term in the formula for the volume of a sphere.

### Cube Root

free Cube Root Calculator

`Y = ∛X`
• X is the value that we are determining the cube root of
• Y is the cube root of X

This is the reverse of cubing a number. A number’s cube root is another number that can be multiplied together three times to make the original number. For example 3 is the cube root of 27, let try it: 3 x 3 x 3, multiplying the first two numbers (3 x 3) results in 9, multiply by 3 again (9 x 3) equals 27, we have multiplied the number by itself three time.

### Cubing

free Cubic Calculator

`Y = X3`
• X is the value to be cubed
• Y is the result of cubing X

To cube a number multiply is by itself three times. A shorthand notation for multiplying a number by itself is to use an exponent, also referred to as raising a number «to the power of…». In the case of cubing we are always raising a number to the power of three, written as X3. Understanding the concept of cubing is key to understanding how to calculate the volume of a cube.

www.volumeofcube.com

## Volume Formulas

(pi = = 3.141592…)

#### Be careful!! Units count. Use the same units for all measurements. Examples

cube = a 3

rectangular prism = a b c

irregular prism = b h

cylinder = b h = pi r 2 h

pyramid = (1/3) b h

cone = (1/3) b h = 1/3 pi r 2 h

sphere = (4/3) pi r 3

ellipsoid = (4/3) pi r1 r2 r3

Units

Volume is measured in «cubic» units. The volume
of a figure is the number of cubes required to fill it completely, like
blocks in a box.

Volume of a cube = side times side times side. Since
each side of a square is the same, it can simply be the length of one
side cubed.

If a square has one side of 4 inches, the volume would
be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic
inches can also be written in3.)

Be sure to use the same units for all measurements.
You cannot multiply feet times inches times yards, it doesn’t make
a perfectly cubed measurement.

The volume of a rectangular prism is the length on
the side times the width times the height. If the width is 4 inches, the
length is 1 foot and the height is 3 feet, what is the volume?

NOT CORRECT …. 4 times 1 times 3 = 12

CORRECT…. 4 inches is the same as 1/3 feet.
Volume is 1/3 feet times 1 foot times 3 feet = 1 cubic foot (or 1 cu.
ft., or 1 ft3).

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## How to find the volume of a cube Master the 7 pillars of school success that I have learned from 25 years of teaching.

### Finding the Volume of a Cube

cube is a three dimensional shape that has six equal sides. The volume is found by using the following formula

The Base Area  of a cube = Length * Width

Example problem from video

Find the volume of cube 1 that has

sides of 5 units.

Step 1 Find the base area

Length * Width

5 * 5 = 25

Step 2 Multiple base area * height

25 * 5 = 125 units^3 Hi welcome to MooMooMath. Today we are going to look at the volume of a cube. Over here we have a cube and a cube is like the Rubik’s cube, it has 6 side, 6 faces, and 4 sides on the base and they are all equal. This cube has sides of 5. This is a prism so we will look at the prism formula and simplify it down to make it specific to the cube. The base area of a prism times the height gives us the volume, so we need to find the area of the base. The base is just a square of 5 so 5 times 5 equals a base area of 25. So I will plug a twenty five in four base area. I will multiple it by the height of the cube, which is 5, so 5 (the height) times 25(the area of a base) is 125 and since this is volume it is units cubed because it is three dimensional. Now let’s simplify this down because we do have a special case. We have a cube that all sides are the same length. Since we have to find the base area and the height and they are all the same we can simplify that to S times S for the base area times S for the height or S cubed. So that is how you find the volume of a cube. So let’s review the rules for the volume of a cube it will be S times S times S or S cubed the units will also be cubed so let’s add that .It is derived from the prism formula which is base area time’s height to get the volume. The base area of a square is S times S times the height which is also S gives us the volume of a cube. Hope this video was helpful in finding the volume of a cube.

Because all the sides of a cube are equal you can also find the volume of a cube by cubing one side.

Formula for volume of a cube = Side^3

5^3 = 125 units ^3

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​Problem 2. This problem is slightly more challenging.

Find the volume of a cube with a diagonal length of 6 units.

Step 3.   Because you have the hypotenuse length you can find the side length                  by dividing hypotenuse/ √2 = 6/√2

Step 4.   Rationalize. =

Step 5a. Plug the side length into the volume formula «side^3»  which                                 becomes (3√2 )^3 = 3*3*3 *√2*√2*√2

Step 5b.  3*3*3 *√2*√2*√2 = 27*2√2

Step 6.    Volume of Cube   27*2√2 = 54√(2 ) units^3

​Step 2. Use your 45-45-90 triangle rules to find the length of one side.

Step 1. The diagonal cuts the cube into two 45 degree angles.

The length of the hypotenuse equals leg length√2

The length of one leg of a 45-45-90 triangle equals hypotenuse/√2

#### Volume Formula for a cube equals: Base Area * Height | Ba * h

Related Sites..

Cubes/Illuminations   Use this interactive cube creator to help understand the volume of a cube, You can also calculate the volume of different cubes, and it will check your work.

www.moomoomath.com

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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

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V=S3

(3a^3b^5)^3

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