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

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

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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.

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:

Calculate the volume of the orthohedron.

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.

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.

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

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.

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.

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

Factoring the $2$

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

where:

$S=$ Surface area

$L=$ length

$W=$ width

$H=$ height

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

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

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

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