So far we have worked with common two-dimensional figures such as the square or the triangle. Three-dimensional figures are those that extend into the third dimension, meaning that in addition to length and width, they also have height (that is, the figure has depth).
What differences do three-dimensional figures have?
Three-dimensional figures have several definitions that we will see next: Below is a three-dimensional figure that we will use to learn each definition - The cube:
Face: it is the flat side of a three-dimensional figure In the cube we have here, there are 6 faces (one of them is painted gray) Edge: these are the lines that connect one face to another in a three-dimensional figure In the cube we have here, there are 12 edges (painted green) Vertex: it is the point that connects the edges In the cube we have here, there are 8 vertices (painted orange)
Volume: it is the amount of space contained within a three-dimensional figure. The units of measurement are cm3 .
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 #2
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
We input the known data into the formula:
2*(3*2+2*5+3*5)
2*(6+10+15)
2*31 = 62
Answer
62
Exercise #3
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 #4
Look at the cuboid below.
What is its surface area?
Video Solution
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
Answer
150
Exercise #5
An unfolded cuboid is shown below.
What is the surface area of the cuboid?
Video Solution
Step-by-Step Solution
To calculate the surface area of the rectangular prism, we will need to identify its three faces (each face appears twice):
1*3
1*8
3*8
The formula for the surface area of a rectangular prism is the sum of all the areas of the faces, that is:
We replace the data in the formula:
2*(1*3+1*8+3*8)= 2*(3+8+24) = 2*35 =
70
And this is the solution!
Answer
70
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
Identify the correct 2D pattern of the given cuboid:
Incorrect
Correct Answer:
Question 3
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 4
Which dimensions may represent a cuboid?
Incorrect
Correct Answer:
All of the above.
Question 5
Given the cuboid of the figure:
Given: volume of the cuboid is 45
What is the value of X?
Incorrect
Correct Answer:
4.5
Exercise #6
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 #7
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 #8
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:
Video Solution
Step-by-Step Solution
The formula to calculate the volume of a cuboid is:
height*length*width
We replace the data in the formula:
3*5*7
7*5 = 35
35*3 = 105
Answer
105 cm³
Exercise #9
Which dimensions may represent a cuboid?
Step-by-Step Solution
There is no limitation or rule regarding the dimensions that a cuboid can have.
Therefore the correct answer is D.
Answer
All of the above.
Exercise #10
Given the cuboid of the figure:
Given: volume of the cuboid is 45
What is the value of X?
Video Solution
Step-by-Step Solution
Volume formula for a rectangular prism:
Volume = length X width X height
Therefore, first we will place the data we are given into the formula:
45 = 2.5*4*X
We divide both sides of the equation by 2.5:
18=4*X
And now we divide both sides of the equation by 4:
4.5 = X
Answer
4.5
Question 1
Look at the following orthohedron:
The volume of the orthohedron is \( 80~cm^3 \).
The length of the lateral edge is 4 meters.
What is the area of the base of the orthohedron? (shaded orange in the diagram)
Incorrect
Correct Answer:
20 cm²
Question 2
Given the cuboid of the figure:
The area of the base of the cuboid is 15 cm²,
The length of the lateral edge is 3 cm.
what is the volume of the cuboid
Incorrect
Correct Answer:
45 cm²
Question 3
Look at the cuboid of the figure.
Its surface area is 122 cm².
What is the width of the cuboid?
Incorrect
Correct Answer:
4 cm
Question 4
Given that the volume of the cuboid is equal to 72 cm³
The length of the cuboid is equal to 6 cm and the height is equal to half the length.
Calculate the surface of the cuboid
Incorrect
Correct Answer:
108 cm²
Question 5
A building is 21 meters high, 15 meters long, and 14+30X meters wide.
Express its volume in terms of X.
Incorrect
Correct Answer:
\( 4410+9450x \)
Exercise #11
Look at the following orthohedron:
The volume of the orthohedron is 80cm3.
The length of the lateral edge is 4 meters.
What is the area of the base of the orthohedron? (shaded orange in the diagram)
Video Solution
Step-by-Step Solution
The formula for the volume of a box is height*length*width
In the specific question, we are given the volume and the height,
and we are looking for the area of the base,
As you will remember, the area is length * width
If we replace all the data in the formula, we see that:
4 * the area of the base = 80
Therefore, if we divide by 4 we see that
Area of the base = 20
Answer
20 cm²
Exercise #12
Given the cuboid of the figure:
The area of the base of the cuboid is 15 cm²,
The length of the lateral edge is 3 cm.
what is the volume of the cuboid
Video Solution
Step-by-Step Solution
To calculate the volume of a cuboid, as we mentioned, we need the length, width, and height.
It is important to note that in the exercise we are given the height and the base area of the cuboid.
The base area is actually the area multiplied by the length. That is, it is the data that contains the two pieces of information we are missing.
Therefore, we can calculate the area by height * base area
15*3 = 45
This is the solution!
Answer
45 cm²
Exercise #13
Look at the cuboid of the figure.
Its surface area is 122 cm².
What is the width of the cuboid?
Video Solution
Step-by-Step Solution
To solve the problem, let's recall the formula for calculating the surface area of a cube:
(width*length + height*width + height*length) *2
Let's substitute the known values into the formula, labelling the missing side X:
2*(3*7+7*X+3*X) = 122
2*(21+7x+3x) = 122
2(21+10x) = 122
Let's now expand the parentheses:
42+20x=122
Now we move terms:
20x=122-42
20x=80
Finally, simplify:
x=4
And that's the solution!
Answer
4 cm
Exercise #14
Given that the volume of the cuboid is equal to 72 cm³
The length of the cuboid is equal to 6 cm and the height is equal to half the length.
Calculate the surface of the cuboid
Video Solution
Step-by-Step Solution
The first step is to calculate the relevant data for all the components of the box.
The length of the box = 6
Given that the height of a cuboid is equal to half its length we are able to deduce the height of the box as follows : 6/2= 3
Hence the height = 3
In order to determine the width, we insert the known data into the formula for the volume of the box:
height*length*width = volume of the cuboid.
3*6*width = 72
18*width=72
We divide by 18:
Hence the width = 4
We are now able to return to the initial question regarding the surface of the cuboid.
Remember that the formula for the surface area is:
(height*length+height*width+length*width)*2
We insert the known data leaving us with the following result:
(3*6+4*3+4*6)*2=
(12+24+18)*2=
(54)*2=
108
Answer
108 cm²
Exercise #15
A building is 21 meters high, 15 meters long, and 14+30X meters wide.
Express its volume in terms of X.
Step-by-Step Solution
We use a formula to calculate the volume: height times width times length.
We rewrite the exercise using the existing data:
21×(14+30x)×15=
We use the distributive property to simplify the parentheses.
We multiply 21 by each of the terms in parentheses:
(21×14+21×30x)×15=
We solve the multiplication exercise in parentheses:
(294+630x)×15=
We use the distributive property again.
We multiply 15 by each of the terms in parentheses:
294×15+630x×15=
We solve each of the exercises in parentheses to find the volume: