# How to calculate the volume of a rectangular prism (orthohedron)

🏆Practice volume of a orthohedron

Students start learning mathematics as early as elementary school, and as they progress, the subject becomes more and more complicated. Among others, the syllabus devotes a part to geometry and requires students to master different shapes and know how to calculate their area and volume. Are you also studying these days how to calculate the volume of a rectangular prism?

## Test yourself on volume of a orthohedron!

Look at the cuboid below:

What is the volume of the cuboid?

Let's see an example:
In this case, we are given a rectangular prism with the following characteristics:

• Length $=6$
• Width $=8$
• Height = $=4$

Therefore, the volume of the rectangular prism is: $192=4\times8\times6$

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

For example:

The question that may appear on the exam may be.

Calculate the volume of:

• Rectangular prism
• Orthohedron
• Cube

The shapes mentioned above, in some problems can be found with less formal names, since we may be asked to calculate the volume of boxes, containers or containers.

It is important to remember that the rectangular prism is a geometric shape with $6$ faces, $12$ Edges and 8 Vertices.

And therefore the calculation of its volume is the same.

If you are curious and interested in the subject, you can learn how to calculate the volume of a prism in the following article: The volume of a right triangular prism.

## Practical exercises for calculating the volume of an orthoctahedron

### Exercise 1

Since the surface area of the rectangular prism is equal to $94cm^2$

The length of the prism is equal to$5 cm$.

The width of the prism is equal to$4 cm$.

What is the volume of the prism?

Solution:

The formula to calculate the surface area of the prism:

Surface area= 2*{(length a * width b) + (height c * length b) + (height c * width a)}.

Solution:

Surface area $= 94cm^2$

Length $= 5cm$

Width $= 4cm$

Height = ?(Unknown $X$)

$94=2\times((5\times 4)+(5\times X)+(4\times X))$

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

$47=20+9X$

$9X=27$

$X=3$

The height is equal to $3cm$

The formula for rectangular prism $=Altura\times Ancho\times Largo=3\times4\times5=60cm³$

Answer is $60cm³$

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### Exercise 2

Given that:

The width of the prism is equal to $8 cm$.

The height of the prism is equal to $4 cm$.

The volume of the prism is equal to $96 cm³$.

What is the length of the prism?

Solution:

We know that:

Width $= 8 cm$

Height $= 4 cm$

Length $= ?$

Prism volume $= 96 cm³$

La fórmula para calcular el volumen del prisma:

Volume of the prism = Height * Width * Length

$96=4\times 8\times(Largo=l)$

$32\times l=96$

we divide by $32$

$l=3$

Correct answer : $l (Largo) = 3 cm$

### Exercise 3

Since the large prism is composed of $4$ prisms of equal size.

The length of the large prism is equal to $10cm$

And its width is equal to half of its length.

Height of the prism is equal to$\frac{4}{5}$ of the length of the prism.

Calculate the volume of one of the small prisms.

Solution:

Given that:

The length of the large prism = The length of the small prism. $= 10cm$

The width of the large prism = The width of the small prism $= 5cm$ (We know this because the width is equal to half the length.

Therefore:

the width is equal $\frac{10}{2}=5cm$

The height of the large prism = $8cm$ (Since the height $\frac{4}{5}$ of the length of the prism, therefore: Height $=10\times\frac{4}{5}=8cm$ )

The height of the small prism = $\frac{8}{4} = 2cm$

We know this from the fact that the height is equal to $4$ lengths of the small prisms.

We substitute in the formula of the volume of the prism = $10\times 5\times 2 = 100$

Volume of the small prism is equal to $100cm³$

Answer is $100$ cm³

Do you know what the answer is?

### Exercise 4

Given that:

The height of the prism is equal to $4cm$.

The width of the height is equal to $X$

The length of the prism is greater by two than the width.

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

Calculate the length of the prism.

Solution:

We put the data into the formula to find the volume of the prism.

Height: $4cm$

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

Width $= X$

Volume $= 12X$

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

$4X^2+8X=12X$

$4X^2=4X$

We divide by $4X$:

$X=1$

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

Correct answer is $3cm$

### Exercise 5

Given a box with a volume of $542 cm³$

Its length is equal to $Ycm$

And its width is twice the length of the box.

The height of the box is $3$ times greater than its length.

Calculate the length of the box. $(Y)$

Solution:

We substitute the data into the formula for the volume of the rectangular prism-shaped box.

Given:

Volume of the box $= 542 cm³$

Length = $Y cm$

Width $= 2 Y$ (because we know that the width of the box is twice the length of the box. That is, $2\times Y$)

Height $= 3 Y$ (because we are given that the height is 3 times the length of the box. That is, $3\times Y$)

$542=Y\times 2Y\times 3Y$

$542=6Y^3$

We divide by $6$ and get cubic raid on both sides of the equation.

$Y=4.48$

The length of the box is equal to. $4.48cm$

## Short and simple exercises

Calculate the volume of the following orthohedron:

• Length $=2$
• Width $=4$
• Height = $=6$

Therefore, the volume of Volume of the orthohedron is: $48=6\times4\times2$

Calculate the volume of the following cube:

• Length $=3$
• Width $=5$
• Height = $=7$

Therefore, the volume of Volume of the cube is: $105=3\times5\times7$

Calculate the volume of the following rectangular prism.

• Length $=7$
• Width $=9$
• Height = $=2$

Therefore, the volume of Volume of the rectangular prism is: $126=7\times9\times2$

Calculate the volume of the following rectangular prism.

• Length $=15$
• Width $=20$
• Height = $=8$

Therefore, the volume of Volume of the rectangular prism is: $2400=15\times20\times8$

Calculate the volume of the following rectangular prism.

• Length $=1$
• Width $=1$
• Height = $=2$

Therefore, the volume of Volume of the rectangular prism is: $2=1\times1\times2$

### Slightly different exercise for calculating the volume of a rectangular prism

Given a rectangular prism with a volume of $48 cm³$.

The length of the base of the box is $2cm$, the width is $4cm$.

What is the height of the rectangular prism?

We use the formula given above:

Length × width × height$= 48 cm³$.

We already know what the length and width are, so we must find the height, which we will denote by the letter $h$.

Thus, we obtain the following equation:

$h×4×2=48$

$h×8=48$

We divide both sides by $8$:

$h=\frac{48}{8}$

$h=6$

As we see, the height of the rectangular prism is $6cm$.

Do you think you will be able to solve it?

## Questions on the subject

### What is the volume of a rectangular prism?

To calculate the volume of a rectangular prism we multiply its length by its width and finally by its height.

### How is the volume of a prism calculated?

To calculate the volume of a rectangular prism we multiply the area of the base by its height.

### What is the formula for calculating the volume of a rectangular prism?

$V=Largo\times Ancho\times Altura$

### How is the volume of a triangular prism calculated?

We calculate the area of the base (Area of the triangle) and multiply it by its height.

Do you know what the answer is?

## Is it useful to memorize the formula for calculating the volume of an orthohedron?

There are cases in which the teachers give the students a sheet with all the formulas, but in other cases the students have to memorize them. Therefore, our recommendation is that you memorize all the formulas little by little. That feeling of stress that invades many when taking an exam can be reduced if you know the formulas by heart. Why?

• Because you have data that you know and can use.
• Because you do not depend on any formula sheet or the professor's willingness to refresh your memory or not.
• Because remembering the formula gives you confidence.

## Review phase: why did you do poorly on the exam?

It is true that a bad grade in mathematics is nothing to be happy about. However, it is important to remember that bad grades are for learning. After your test has been returned to you, invest some time to review the test. It is vital to note that sometimes it will not be easy to do this alone and so it may be helpful to do this with a math tutor. The purpose of this review is to understand what hindered your path to a good grade so that you can focus on it and do better in the future. So where did you fail this time?

• You lacked concentration or went blank due to nerves or stress. Solution: do breathing exercises and meditation before taking the exam.
• The exercises were too complicated. This is one of the most common reasons for failing the exam. Solution: practice exercises of different levels before the exam. Many times, we practice only easy level exercises to feel more confident and it is a complete mistake.
• You have been deducted points for calculation errors. Small miscalculations can affect your mood. Solution: check your answers several times before you give the test to the teacher.
• You didn't have time to answer all the exercises. Learning to manage your time during an exam is paramount. When you know how to manage it properly, you will have the power in your hands. Solution: study for the exam with a stopwatch to improve your times.

A grade does not represent you, it is only a small part of you. Mathematics is a subject in which you can improve with perseverance and prove it in future exams. Never forget this: the advantage of getting a bad grade (everything is relative, of course) is that you can always improve.

## Preparing for the exam with friends: does it really pay off?

There is a direct correlation between the study before an exam and the grade you get. Many times, when we try to understand why we didn't get the results we expected, we realize that the problem was in the preparation and study beforehand. Usually, these are students who are good friends and who decided to study together in a group. Sometimes these study groups can go well, but sometimes they don't allow you to concentrate for long periods of time or they become more like social gatherings than study groups. Here are some questions that can help you decide if studying in a group with friends is effective for you or not:

• Did you manage to understand the gaps you had with the help of your friends?
• Did you have to continue studying at home after studying with your friends?
• How many exercises did you manage to solve while studying in a group?
• Did you feel you were prepared for the exam?
• A good hint: what grade did the friends you studied with get?
Do you think you will be able to solve it?

## Don't get left behind: pick up your pace with a private teacher

Since we are talking about formulas, you should know the following: the fewer gaps you have in mathematics, the easier it will be for you to get a good grade in mathematics. It is common knowledge that in the more advanced groups (especially in junior high and high school) there are many more tests and quizzes.

Sometimes, a private lesson can help you make a lot of progress and engrave the formulas in your head and make you master them. The orthoctahedron is one more shape that you will have to work on in exams in a myriad of ways. The best way to memorize the formula is to practice non-stop problems related to the orthohedron, with special attention to those that require you to calculate its volume.

## examples with solutions for volume of a orthohedron

### Exercise #1

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

### Exercise #2

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 #3

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 #4

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

### Exercise #5

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:

$4,410+9,450x$

$4410+9450x$

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