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

🏆Practice surface area of a orthohedron

Rectangular Prisms are made up of $6$ different rectangles. When faced with an exercise or exam that asks you to calculate the surface area of a rectangular Prism, use the formula below.

## The formula: how to calculate the area of a rectangular prism (rectangular orthohedron)?

$S=2 \times (Ancho \times Largo + Altura \times Ancho + Altura \times Largo)$

S= surface area

## Test yourself on surface area of a orthohedron!

A cuboid is shown below:

What is the surface area of the cuboid?

If we take as an example an orthohedron with the following characteristics, its surface area will be calculated as follows:

Width = $5$ cm

Length = $2$ cm

Height = $3$ cm

Now, we apply the formula:

$S= 2×(5×2+3×5+3×2)=?$

Thus, by solving the exercise we will obtain that the surface area of the rectangular prism (orthohedron) is $62$ cm².

If this exercise is easy for you and you are interested in learning how to calculate the surface area of a prism, you can learn it in the following article: Surface area of triangular prisms.

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

For example: Rectangular Prism, Orthohedron and Cube.

Soit is important to remember that it is a geometric shape with $6$ faces, $12$ Edges and $8$ Vertices.

What is our conclusion?

That the surface area of a rectangular prism (orthohedron) is the sum of the areas of all the rectangles that form it.

Throughout primary and secondary school, you will have to deal with exercises of all kinds related to the field of geometry. So you will need to know how to calculate the surface area of a rectangular prism. We present you the formula that will help you to do it and give you some tips to internalize the learned materials in a better way.

If we take as an example a rectangular prism with the following characteristics, its surface area will be calculated as follows:

Width $=2$ cm

Length $=4$ cm

Height $=3$ cm

The surface area of the rectangular prism is:

$S= 2×(2×4+3×2+3×4)=52$

A that the surface area of an orthohedron is the sum of the areas of all the rectangles that form it. Let's see it illustrated in the following picture:

Answer:

Thus, by solving the exercise we will obtain that the surface area of the rectangular prism is $52$ cm².

What is our conclusion?

If you are interested in this article you may also be interested in the following articles:

Orthohedron - rectangular prism

The cube

For a wide range of math articles visit Tutorela's blog.

## Exercises to calculate the surface of a rectangular prism (orthohedron)

### Exercise 1

Given two orthohedra

Task:

Are the surfaces of the two orthohedra the same or different?

Solution:

Let's observe that the orthohedra are identical, they are just presented differently.

If we turn one of them upside down, it will be clear that the cubes are identical.

We can verify by calculus.

Right orthohedron :

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

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

$4+6+12=$

$22$

Left Orthohedron :

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

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

$4+6+12=$

$22$

Answer:

The surfaces are equal.

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Test your knowledge

### Exercise 2

Given that the area of the orthohedron is equal to $94$ cm².

The height of the orthohedron is equal to $5$ cm and the width is $4$ cm.

Calculate the volume of the orthohedron

Task:

Solution:

Area = $94$ cm²

Length = $?$ cm

Width = $4$ cm

Height = $5$

Replace the height by $X$

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

$47=20+9X$

$9X=27$

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

We replace it in the volume formula:

$5\times4\times3=60$

Answer:

The volume of the orthohedron is equal to $60$ cm³.

### Exercise 3

Given a cube with the following information:

Width $=8$ cm

Length $=14$ cm

Height $=3$ cm

How to calculate the surface area of the cube.

$S= 2\times(8×14+3×8+3×14)=356$

Answer:

The surface area of the cube is: $356$ cm².

Do you know what the answer is?

### Exercise 4

Given a rectangular prism with the following information:

Width $=5$ cm

Length $=3$ cm

Height $=7$ cm

How to calculate the surface area of the rectangular prism.

$S= 2\times(5×3+7×5+7×3)=142$

Answer:

$142$ cm²

### Exercise 5

Given a rectangular prism with the following information:

Width $=16$ cm

Length $=12$ cm

Height $=19$ cm

How to calculate the surface area of the rectangular prism.

$S= 2\times(16×12+19×16+19×12)=1448$

Answer:

The surface area of the rectangular prism is: $1448$ cm².

Check your understanding

## Review questions

### How do you calculate the total area of a rectangular prism?

To calculate the total area of a rectangular prism, we have to calculate the areas of each of its faces (6 faces) and then add the area of all of them to obtain the total area.

### What is the formula for finding the area of a prism?

$S=2LW+2LH+2WH$

Factoring the $2$

$S=2(LW+LH+WH)$

where:

$S=$ Surface area

$L=$ length

$W=$ width

$H=$ height

Do you think you will be able to solve it?

### How to calculate the area and volume of a rectangular prism?

The area of a rectangular prism is calculated with the formula:

$S=2(LW+LH+WH)$

While the volume is calculated with the following formula:

$V=L\times w\times h$

Example

Let the following rectangular prism have the following dimensions

Width $=7\operatorname{cm}$

Length $=3\operatorname{cm}$

Height $=5\operatorname{cm}$

Let's calculate the area with the formula

$S=2(3\times7+3\times5+7\times5)=$

$S=2(21+15+35)=$

$S=2(71)=142\operatorname{cm}^2$

Now we calculate the volume

$V=L\times w\times h$

$V=3\operatorname{cm}\times7\operatorname{cm}\times5\operatorname{cm}=105\operatorname{cm}^3$

Result

$S=142\operatorname{cm}^2$

$V=105\operatorname{cm}^3$

### How do you calculate the area of a rectangular box?

To calculate the area of a rectangular box we add the areas of its six faces, or by using the following formula:

$S=2(LW+LH+WH)$

Where:

$S=$ Surface area

$L=$ length

$W=$ width

$H=$ height

Test your knowledge

## examples with solutions for surface area of a 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

### Answer

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

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

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