# Orthohedron - Examples, Exercises and Solutions

## 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.

## 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

105 cm³

## examples with solutions for orthohedron

### Exercise #1

An unfolded cuboid is shown below.

What is the surface area of the cuboid?

### Step-by-Step Solution

To calculate the surface area of the rectangular prism, we will need to identify its three faces (each face appears twice):

1*3

1*8

3*8

The formula for the surface area of a rectangular prism is the sum of all the areas of the faces, that is:

We replace the data in the formula:

2*(1*3+1*8+3*8)=
2*(3+8+24) =
2*35 =

70

And this is the solution!

70

### Exercise #2

Look at the cuboid below.

What is its surface area?

### Step-by-Step Solution

We identified that the faces are

3*3, 3*11, 11*3
As the opposite faces of an cuboid are equal, we know that for each face we find there is another face, therefore:

3*3, 3*11, 11*3

or

(3*3, 3*11, 11*3 ) *2

To find the surface area, we will have to add up all these areas, therefore:

(3*3+3*11+11*3 )*2

And this is actually the formula for the surface area!

We calculate:

(9+33+33)*2

(75)*2

150

150

### Exercise #3

Given the cuboid of the figure:

Given: volume of the cuboid is 45

What is the value of X?

### Step-by-Step Solution

Volume formula for a rectangular prism:

Volume = length X width X height

Therefore, first we will place the data we are given into the formula:

45 = 2.5*4*X

We divide both sides of the equation by 2.5:

18=4*X

And now we divide both sides of the equation by 4:

4.5 = X

4.5

### Exercise #4

Look at the following orthohedron:

The volume of the orthohedron is $80~cm^3$.

The length of the lateral edge is 4 meters.

What is the area of the base of the orthohedron?

### Step-by-Step Solution

The formula for the volume of a box is height*length*width

In the specific question, we are given the volume and the height,

and we are looking for the area of the base,

As you will remember, the area is length * width

If we replace all the data in the formula, we see that:

4 * the area of the base = 80

Therefore, if we divide by 4 we see that

Area of the base = 20

20 cm²

### Exercise #5

Given the cuboid of the figure:

The area of the base of the cuboid is 15 cm²,

The length of the lateral edge is 3 cm.

what is the volume of the cuboid

### Step-by-Step Solution

To calculate the volume of a cuboid, as we mentioned, we need the length, width, and height.

It is important to note that in the exercise we are given the height and the base area of the cuboid.

The base area is actually the area multiplied by the length. That is, it is the data that contains the two pieces of information we are missing.

Therefore, we can calculate the area by height * base area

15*3 = 45

This is the solution!

45 cm²

## examples with solutions for orthohedron

### Exercise #1

Given the volume of the cuboid equal to 72 cm³

Length of the cuboid is equal to 6 cm and the height is equal to half the length.

Calculate the surface of the cuboid

### Step-by-Step Solution

First, let's start by figuring out the data for all the components of the box.

Given that the length is - 6

It is known that the height is equal to half the length. - 6/2= 3
Height = 3

To find the width, we will place the data we have in the formula for the volume of the box:

height*length*width = volume of the rectangular prism

We replace and reveal that:

3*6*width = 72

18*width=72

We divide by 18:

Width = 4

Now we can move on to find what was asked in the question,

Remember that the formula for the surface area is:

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

We place the data we know:

(3*6+4*3+4*6)*2=

(12+24+18)*2=

(54)*2=

108

108 cm²

### Exercise #2

An unfolded cuboid is shown below.

What is the surface area of the cuboid?

### Step-by-Step Solution

To calculate the surface area of the rectangular prism, we will need to identify its three faces (each face appears twice):

1*3

1*8

3*8

The formula for the surface area of a rectangular prism is the sum of all the areas of the faces, that is:

We replace the data in the formula:

2*(1*3+1*8+3*8)=
2*(3+8+24) =
2*35 =

70

And this is the solution!

70

### Exercise #3

Given the volume of the cuboid equal to 72 cm³

Length of the cuboid is equal to 6 cm and the height is equal to half the length.

Calculate the surface of the cuboid

### Step-by-Step Solution

First, let's start by figuring out the data for all the components of the box.

Given that the length is - 6

It is known that the height is equal to half the length. - 6/2= 3
Height = 3

To find the width, we will place the data we have in the formula for the volume of the box:

height*length*width = volume of the rectangular prism

We replace and reveal that:

3*6*width = 72

18*width=72

We divide by 18:

Width = 4

Now we can move on to find what was asked in the question,

Remember that the formula for the surface area is:

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

We place the data we know:

(3*6+4*3+4*6)*2=

(12+24+18)*2=

(54)*2=

108

108 cm²

### Exercise #4

A building is 21 meters high, 15 meters long, and 14+30X meters wide.

Express its volume in terms of X.

### Step-by-Step Solution

We use a formula to calculate the volume: height times width times length.

We rewrite the exercise using the existing data:

$21\times(14+30x)\times15=$

We use the distributive property to simplify the parentheses.

We multiply 21 by each of the terms in parentheses:

$(21\times14+21\times30x)\times15=$

We solve the multiplication exercise in parentheses:

$(294+630x)\times15=$

We use the distributive property again.

We multiply 15 by each of the terms in parentheses:

$294\times15+630x\times15=$

We solve each of the exercises in parentheses to find the volume:

$4,410+9,450x$

$4410+9450x$

### Exercise #5

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

The height of the prism is 12 units.

Calculate its volume.