# Cubes

🏆Practice cube

A cube is a type of cuboid in which all three dimensions (length, width and height) are identical. All cubes are made up of of six identical squares.

To find the volume of a cube we must go through the same steps as to find the volume of an cuboid, that is:

Length (L) × Depth (W) × Height (H).

Since the length, width and height are all equal, we only need to know one of them to calculate the volume.

## Test yourself on cube!

A cube has a total of 14 edges.

## Example exercise: volume and surface area of a cube

We have a cube whose length is $2$ cm and we are asked to find its volume and surface area.

## Finding the volume of a cube

The volume of a cube is equal to length × width × height.

Since the length, width and height of a cube are all equal, in our case the width and height of our given cube will also be $2$ cm. Therefore,

$8=2\times2\times2$

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## Finding the surface area of a cube

To find the total surface area of a cube, we will first find the surface area of one of its faces and then multiply the result by 6 (remember that cubes are composed of six identical square faces).

The area of each square is
$4=2\times2$

Therefore, the surface area of the cube will be:

$4×6=24\operatorname{cm}$

How to calculate the area of an orthohedron - rectangular prism or cube.

How to Calculate the Volume of a Rectangular Prism (Orthohedron)

Orthohedron - rectangular prism

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## Example exercises

### Example exercise 1

Given that:

The length of each side of the given cube is equal to $3$ cm.

Question:

What is the volume of the cube?

Solution:

The volume of a cube (and the volume of a cuboid) is equal to:

Length × Width × Height

Therefore the volume of the cube: $=3^3=27$

$27~cm³$

Do you know what the answer is?

### Example exercise 2

Given that:

Given a cube in which each face has a surface area of $6$ cm.

Assignment:

What is the total surface area of the cube?

Solution:

The total surface area of the cube is the combined area of all of its faces, ie:

Face area

$6\times6=36$

$36~cm²$

### Example exercise 3

Given that:

In the given cube, the length of each edge is equal to $3$ cm.

Question:

What is the length of the diagonal of the face?

Solution:

To solve this question we will use the Pythagorean Theorem to find the length of the diagonal of the face:

$A^2+B^2=C^2$

Or, in our case:

$Edge^2+Edge^2=Diagonal^2$

$=3^2+3^2$

$=18$

$\sqrt{18}=3\times\sqrt{2}=diagonal$

$3\sqrt{2}$

### Example exercise 4

Given a cube whose edge length is equal to $5$ cm.

Find the volume of the cube.

Solution:

The volume of the cube is equal to the length of the face of the cube to the power of $3$

We can write it like this:

$5^3=125$

$125~cm³$

### Example exercise 5

Given a cube whose volume is equal to $112$ cm³

Question:

How many whole cubes with a volume of $10$ cm³ can fit inside the given cube?

Solution:

We divide the volume of the large cube into $10$ to find out how many cubes of $10$ cm³ fit into the given cube:

$\frac{112}{10}=11\frac{1}{5}$

Since we are only asked about whole cubes, it is possible to enter $11$ cubes into the cube whose volume is $112$ cm³.

$11$ cubes.

Do you think you will be able to solve it?

## Review questions

### What is a cube?

A cube is a cuboid with six square, equal faces (all the sides are equal).

### How do we find the surface area of a cube?

To find the total surface area of a cube, all we need is the value of one of its sides (since all sides are equal).

Then, we find the surface area of one face by multiplying the side to the power of three.

Lastly, we multiply the surface area of one face by six (since cubes have six equal sides).

#### Example exercise

Task. Find the total surface area of the following given cube, which has a side length of $7\operatorname{cm}$

Solution:

Let's start by finding the area of just one face:

$Area=7\operatorname{cm}\times7\operatorname{cm}=49\operatorname{cm^2}$

Now, let's multiply the area of one face by six to find the total surface area:

$49\operatorname{cm^2}\times6=294\operatorname{cm^2}$

$=294 \operatorname{cm^2}$

### What is the formula used to find the volume of a cube?

The find the volume of a cube, we multiply its three sides.

Remember: since each face is square, all its sides have the same length.

$=$,$\times$

This formula can also be expressed as:

$V=L^3$

since all the sides are equal.

### Finding the volume of a cube: additional practice

#### Example 1

Task. Find the volume of a cube with a side length of$4\operatorname{cm}$

Solution:

Using our formula, we get:

$V=L^3$

$V=\left(4\operatorname{cm}\right)^3=64\operatorname{cm^3}$

$V=64\operatorname{cm^3}$

#### Example 2

Task. Find the volume of a cube with a side length of $8\operatorname{cm}$

Solution:

Again, we will use our formula to find the volume:

$V=L^3$

$V=\left(8\operatorname{cm}\right)^3=512\operatorname{cm^3}$

$V=512\operatorname{cm^3}$

Do you know what the answer is?

## examples with solutions for cube

### Exercise #1

All faces of the cube must be?

Squares

Find a,b

### Video Solution

$a=b=5$

### Exercise #3

Which of the following figures represents an unfolded cube?

### Exercise #4

A cube has a total of 14 edges.

False.

### Exercise #5

Look at the cube below.

Do all cubes have 6 faces, equaling its surface area?