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.

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

Finding the surface area of the cuboid (without the bases)

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.

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

Do you know what the answer is?

Question 1

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

Total surface area of a cuboid (with all faces and bases)

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 in our day-to-day

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?

Check your understanding

Question 1

Calculate the volume of the rectangular prism below using the data provided.

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.

How many faces does a cuboid have?

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