# Cuboids

🏆Practice orthohedron

## Structure of a rectangular cuboid

The rectangular cuboid, or just cuboid, is a three-dimensional shape that consists of six rectangles. Each rectangle is called a face. Every rectangular cuboid has six faces (The top and bottom faces are often called the top and bottom bases of the rectangular cuboid). It is important to understand that there are actually $3$ pairs of faces, and each face will be identical to its opposite face.

The straight lines formed by two intersecting sides are called edges (or sides). Every cuboid has $12$ edges.

The meeting point between two edges is called the vertex. Each cuboid has $8$ vertices.

## Test yourself on orthohedron!

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

It is important to remember that in an exam the name of the shape may vary from one exercise to another.

• Cuboid
• Rectangular prism
• Orthohedron
• Cube (a special kind of cuboid)
• Rectangular parallelepiped
• Orthogonal parallelepiped

So, it is important to remember that all of these describe a geometric shape with $6$ faces, $12$ Edges and $8$ Vertices.

## The three dimensions of the cuboid

As we know, a cuboid is a three-dimensional shape and therefore each cuboid can be said to have a length, width and height.

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## Finding the volume of a cuboid

The volume of a cuboid can be found by multiplying the three dimensions of the cuboid (i.e. length, width and height).

## Finding the surface area of the cuboid (without the bases)

If needed, we can find the surface area of just the lateral faces of a cuboid (without the bases) by adding together the areas of the four rectangles that "wrap" the cuboid, that is, without the base rectangles.

$S_s=2\left(W\times H+L\times H\right)$

Do you know what the answer is?

## Total surface area of a cuboid (with all faces and bases)

We can find the total surface area of a cuboid by adding the areas of all six rectangles that form the cuboid (i.e., including the bases).

$S=2\left(W\times L+H\times W+H\times L\right)$

Let's use an example to help us understand how to find the surface area:

Given a cuboid whose length is $4$ cm, whose width is $3$ cm and whose height is $5$ cm.

We are asked to find both the volume and the surface area of the cuboid.

Calculate the volume of the cuboid by multiplying the three dimensions. We will receive: $60$ cm³

Let's continue:

Now we will calculate the total surface area of the cuboid by using the areas of the six rectangles.

The areas we will receive are:

$12$ cm², $20$ cm² and $15$ cm².

Now since each face has an opposite face, we will multiply each area by $2$.

$24$ cm², $40$ cm² and $30$ cm².

Lastly, we will add the three values together, and get the total surface area of the cuboid, which will give us $94$ cm².

## Cuboids in our day-to-day

Cuboids are very common shapes in our day-to-day world.

Look around and you will notice that you are surrounded by many objects that have this shape: shoeboxes, smartphones, your favorite cereal box, your bedroom, etc. Learning how to work with this shape will allow you to easily answer questions like:

Is there enough space in the bedroom for a new desk?

Will this box be big enough?

How much paint do I need to paint my house?

Can you think of more examples from your own life?

## Review questions

### Describing a cuboid

A cuboid is a three-dimensional shape formed by three pairs of rectangles, called faces. Each pair of faces are placed opposite each other. The opposite faces are equal.

### How many faces does a cuboid have?

A cuboid has $6$ faces. Two opposite faces can be called bases, and the remaining four are called lateral faces.

Do you think you will be able to solve it?

### Are the opposite faces of a cuboid the same?

Yes! The opposite faces of a cuboid are equal.

### How many different parts make up a cuboid?

A cuboid has $6$ faces (in the form of rectangles), $12$ vertices and $8$ edges.

The cube

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

How to calculate the volume of a rectangular prism (orthoctahedron)

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## Example exercises

### Example exercise 1

Given that:

The length of the cuboid is equal to $7$ cm.

Its width is equal to $3$ cm.

The height of the cuboid is equal to $5$ cm.

Question:

What is the volume of the cuboid?

Solution:

To find the solution, we put the given values into our formula for finding the volume of a cuboid:

$Height\times Width\times Length=Volume~of~cuboid$

$7\times3\times5=105$

$105$ cm²

### Example exercise 2: Using variables

Given that:

The height of the given cuboid is equal to $4\operatorname{cm}$.

The width is equal to $X$.

The length is greater by two than the width.

The volume of the prism is equal to $12X$.

Find the length of the cuboid.

Options:

1. $3$ cm
2. $12$ cm
3. $1$ cm
4. $4$ cm

Solution:

We start by choosing one of the options given and by putting the values into our formula to see if they work:

$Height: 4$ cm

$Length: X+2$ (Because given a length greater by $2$ than the width, i.e. $X+2$)

$Width= X$

$Volume= 12X$

$12X=4\cdot X(X+2)$

$4X^2+8X\text{ }=12X$

We divide by $4X$: / $4X^2=4X$

$X=1$

The length of the cuboid is equal to: $X+2$ therefore $1+2=3\operatorname{cm}$.

The correct answer is $N°1=3\operatorname{cm}$

Do you know what the answer is?

### Example exercise 3

Given that:

The width of the cuboid is equal to $12$ cm.

The length is equal to $40%$ of the width.

The height is equal to $30%$ of the width.

Find the volume of the cuboid.

Solution:

We sort the given values and put them into our formula:

$Width=12$ cm

Since the length is equal to $40%$ of the width:

$\frac{40\cdot12}{100}=4.8$

$Length=4.8$ cm

Since the height is equal to $30%$ of the width:

$\frac{30\times12}{100}=3.6$

$Length=4.8$

Now that we have all of our values, we can continue with our formula:

$12\times4.8\times3.6=$

$207.36\approx207.3$

$207.3cm³$

### Example exercise 4: Using surface area to find volume

Given that the surface area of the cuboid is equal to $94$ cm³.

The length is equal to $5$ cm.

The width is equal to $4$ cm.

What is the volume of the cuboid?

Options:

1. $9$ cm³
2. $97$ cm³
3. $60$ cm³
4. $40$ cm³

Remember that the formula to find the surface area of the cuboid is:

$SurfaceArea=2\times{(lengtha\times widthb)+(heightc\times lengthb)+(height~c\times width~a)}$

Solution:

Area = $94$ cm³

Length = $5$ cm

Width = $4$ cm

Height = $?$ (Unknown $X$)

$94=2\cdot(\frac{20}{(5\cdot4)}+\frac{5X}{(5\cdot X)}+\frac{4X}{(4\cdot X)})$

$47=20+9X$

$9X=27$

$X=3$

Height equals $3$cm

$Volume~of~cuboid=Height\times Width\times Length=3\times4\times5=60~cm³$

$N°3.60\operatorname{cm}³$

### Example exercise 5

Given the following two cuboids:

Question:

Are the surface areas of the two cuboids the same or different?

Solution:

Let's assume that the cuboids are identical, but they are just presented differently.

If we flip one of them on its side so that their dimensions are matching, it will be clear that they are identical.

But to just to be sure, let's verify by using our formula:

Right cuboid:

$2\left(1\times2\right)+2\left(1\times3\right)+2\left(3\times2\right)=$

$2\times2+2\times3+6\times6=$

$4+6+18=$

$28$

Left cuboid:

$2\left(1\times2\right)+2\left(1\times3\right)+2\left(3\times2\right)=$

$2\times2+2\times3+6\times6=$

$4+6+18=$

$28$

The surface areas are equal

### Example exercise 6

The length of a cuboid is equal to $5$ cm and its width is $4$ cm.

Find the volume of the cuboid.

Solution:

Area = $94$ cm³

Length = $4$ cm

Width = $4$ cm

Height = $?$

Replace the height by $X$

$94=2((5\times4)+(5\times X)+(4\times X))$ / :divide into $2$

$47=20+9X$

$9X=27$

$X=3$ The height is equal to $3$ cm.

We replace it in the volume formula:

$5\times4\times3=60$

The volume of the cuboid is equal to $60~cm³$

Do you think you will be able to solve it?

## examples with solutions for orthohedron

### Exercise #1

A cuboid has the dimensions shown in the diagram below.

Which rectangles form the cuboid?

### Step-by-Step Solution

Every cuboid is made up of rectangles. These rectangles are the faces of the cuboid.

As we know that in a rectangle the parallel faces are equal to each other, we can conclude that for each face found there will be two rectangles.

Let's first look at the face painted orange,

It has width and height, 5 and 3, so we already know that they are two rectangles of size 5x6

Now let's look at the side faces, they also have a height of 3, but their width is 6,

And then we understand that there are two more rectangles of 3x6

Now let's look at the top and bottom faces, we see that their dimensions are 5 and 6,

Therefore, there are two more rectangles that are size 5x6

That is, there are
2 rectangles 5X6

2 rectangles 3X5

2 rectangles 6X3

Two 5X6 rectangles

Two 3X5 rectangles

Two 6X3 rectangles

### Exercise #2

Look at the the cuboid below.

What is its surface area?

### Step-by-Step Solution

First, we recall the formula for the surface area of a cuboid:

(width*length + height*width + height*length) *2

As in the cuboid the opposite faces are equal to each other, the given data is sufficient to arrive at a solution.

We replace the data in the formula:

(8*5+3*5+8*3) *2 =

(40+15+24) *2 =

79*2 =

158

158

### Exercise #3

Look at the cuboid below.

What is the surface area of the cuboid?

### Step-by-Step Solution

Let's see what rectangles we have:

8*5

8*12

5*12

Let's review the formula for the surface area of a rectangular prism:

(length X width + length X height + width X height) * 2

Now let's substitute all this into the exercise:

(8*5+12*8+12*5)*2=
(40+96+60)*2=
196*2= 392

This is the solution!

392 cm²

### Exercise #4

A cuboid is shown below:

What is the surface area of the cuboid?

### Step-by-Step Solution

Remember that the formula for the surface area of a cuboid is:

(length X width + length X height + width X height) 2

We input the known data into the formula:

2*(3*2+2*5+3*5)

2*(6+10+15)

2*31 = 62

62

### Exercise #5

Given the cuboid whose length is equal to 7 cm

Width is equal to 3 cm

The height of the cuboid is equal to 5 cm

Calculate the volume of the cube

### Step-by-Step Solution

The formula to calculate the volume of a cuboid is:

height*length*width

We replace the data in the formula:

3*5*7

7*5 = 35

35*3 = 105