Cuboids

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.

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

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

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.

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

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 $2$.

We will receive:

$24$ 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 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?

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.

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

Yes! The opposite faces of a cuboid are equal.

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

If you are found this article helpful, you may also be interested in the following:

The cube

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

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

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

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:

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

$7\times3\times5=105$

Answer:

$105$ cm²

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 $12X$.

Task:

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 $4X$: / $4X^2=4X$

$X=1$

Answer:

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

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

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.

Task:

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:

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

$207.36\approx207.3$

Answer:

$207.3cm³$

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.

Task:

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

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

Correct answer:

$N°3.60\operatorname{cm}³$

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:

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

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

$4+6+18=$

$28$

Answer:

The surface areas are equal

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

Task:

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:

$5\times4\times3=60$

Answer:

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