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.
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, 12Edges and 8Vertices.
The three dimensions of the cuboid
As we know, a cuboid is a three-dimensional shape and therefore each cuboid can be said to have a length, width and height.
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Test your knowledge
Question 1
A cuboid has the dimensions shown in the diagram below.
Which rectangles form the cuboid?
Incorrect
Correct Answer:
Two 5X6 rectangles
Two 3X5 rectangles
Two 6X3 rectangles
Question 2
A cuboid is 9 cm long, 4 cm wide, and 5 cm high.
Calculate the volume of the cube.
Incorrect
Correct Answer:
180 cm³
Question 3
A cuboid is shown below:
What is the surface area of the cuboid?
Incorrect
Correct Answer:
62
Finding the volume of a cuboid
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.
Ss=2(W×H+L×H)
Do you know what the answer is?
Question 1
A rectangular prism has a base measuring 5 units by 8 units.
The height of the prism is 12 units.
Calculate its volume.
Incorrect
Correct Answer:
480
Question 2
Below is a cuboid with a length of
8 cm.
Its width is 2 cm and its height is
4 cm.
Calculate the volume of the cube.
Incorrect
Correct Answer:
64 cm³
Question 3
Calculate the surface area of the orthohedron below using the data in the diagram.
Incorrect
Correct Answer:
62
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(W×L+H×W+H×L)
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 cuboid
If its length is equal to 7 cm:
Its width is equal to 3 cm:
Its height is equal to 5 cm:
Incorrect
Correct Answer:
105 cm³
Question 2
Calculate the volume of the rectangular prism below using the data provided.
Incorrect
Correct Answer:
48
Question 3
Calculate the volume of the rectangular prism below using the data provided.
Incorrect
Correct Answer:
180
Review questions
Describing a cuboid
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.
Do you think you will be able to solve it?
Question 1
Given the cuboid of the figure:
What is its volume?
Incorrect
Correct Answer:
180
Question 2
Look at the cuboid below.
What is the surface area of the cuboid?
Incorrect
Correct Answer:
392 cm²
Question 3
Look at the cuboid below:
What is the volume of the cuboid?
Incorrect
Correct Answer:
480 cm³
Are the opposite faces of a cuboid the same?
Yes! The opposite faces of a cuboid are equal.
How many different parts make up a cuboid?
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:
A rectangular prism has a base measuring 5 units by 8 units.
The height of the prism is 12 units.
Calculate its volume.
Incorrect
Correct Answer:
480
Question 2
Below is a cuboid with a length of
8 cm.
Its width is 2 cm and its height is
4 cm.
Calculate the volume of the cube.
Incorrect
Correct Answer:
64 cm³
Question 3
Calculate the surface area of the orthohedron below using the data in the diagram.
Incorrect
Correct Answer:
62
Example exercise 5
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(1×2)+2(1×3)+2(3×2)=
2×2+2×3+6×6=
4+6+18=
28
Left cuboid:
2(1×2)+2(1×3)+2(3×2)=
2×2+2×3+6×6=
4+6+18=
28
Answer:
The surface areas are equal
Example exercise 6
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((5×4)+(5×X)+(4×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:
5×4×3=60
Answer:
The volume of the cuboid is equal to 60cm3
Do you think you will be able to solve it?
Question 1
Calculate the volume of the cuboid
If its length is equal to 7 cm:
Its width is equal to 3 cm:
Its height is equal to 5 cm:
Incorrect
Correct Answer:
105 cm³
Question 2
Calculate the volume of the rectangular prism below using the data provided.
Incorrect
Correct Answer:
48
Question 3
Calculate the volume of the rectangular prism below using the data provided.
Incorrect
Correct Answer:
180
Examples with solutions for Cuboids
Exercise #1
A cuboid has the dimensions shown in the diagram below.
Which rectangles form the cuboid?
Video Solution
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?
Video Solution
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
Answer
158
Exercise #3
Identify the correct 2D pattern of the given cuboid:
Step-by-Step Solution
Let's go through the options:
A - In this option, we can observe that there are two flaps on the same side.
If we try to turn this net into a box, we should obtain a box where on one side there are two faces one on top of the other while the other side is "open", meaning this net cannot be turned into a complete and full box.
B - This net looks valid at first glance, but we need to verify that it matches the box we want to draw.
In the original box, we see that we have four flaps of size 9*4, and only two flaps of size 4*4, if we look at the net we can see that the situation is reversed, there are four flaps of size 4*4 and two flaps of size 9*4, therefore we can conclude that this net is not suitable.
C - This net at first glance looks valid, it has flaps on both sides so it will close into a box.
Additionally, it matches our drawing - it has four flaps of size 9*4 and two flaps of size 4*4.
Therefore, we can conclude that this net is indeed the correct net.
D - In this net we can see that there are two flaps on the same side, therefore this net will not succeed in becoming a box if we try to create it.
Answer
Exercise #4
Look at the cuboid below.
What is the surface area of the cuboid?
Video Solution
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!
Answer
392 cm²
Exercise #5
A cuboid is shown below:
What is the surface area of the cuboid?
Video Solution
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