Three-Dimensional Figures

Three-dimensional figures

What are three-dimensional figures?

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 $cm^3$ .

Detailed explanation

Start practice

Test yourself on cuboids!

A cuboid is shown below:

What is the surface area of the cuboid?

Practice more now

Cuboid

The cuboid is a three-dimensional figure composed of 6 rectangles.

Each cuboid has:

6 faces: the rectangles that make up the cuboid - three pairs of rectangles that can be different from each other.
12 edges: the edges of the cuboid (divided into length, width, and height) - marked in green
8 vertices: the points where the edges meet - marked in orange

You can learn more about the parts of the cuboid by reading this article

Face diagonals:

The diagonals that go from one vertex to another on the same face as long as the vertices belong to the same face - marked in blue

Diagonal of the cuboid:

The diagonals that go from one vertex to another on different faces, as long as the 2 vertices belong to the same face - marked in red

Volume of the cuboid

The volume of the rectangular prism is ( \times \~the~width~\times \~the~length\~the~height\)

For more information about the volume of the rectangular prism click here

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Test your knowledge

Question 1

Look at the cuboid below.

What is its surface area?

Question 2

Look at the cuboid below.

What is the surface area of the cuboid?

Question 3

Look at the cuboid below:

What is the volume of the cuboid?

Surface area of a cuboid

The formula to find the surface area is(width\times length + height\times width+ height\times length)\times \2

You can read again about the surface area of a rectangular prism by clicking here

Lateral area of a cuboid

It is the sum of the area of the four lateral rectangles (without the bases).
The lateral area of a cuboid can be calculated with the following formula:
a- Length
b- Width
h- Height

2\times \ah+2\times \bh= lateral~area~of~a~cuboid

You can read again about the lateral area of a cuboid by clicking here
For more information about the cuboid click here

Do you know what the answer is?

Question 1

An orthohedron has the dimensions: 4, 7, 10.

How many rectangles is it formed of and what are their dimensions?

Question 2

A cuboid has a length of is 9 cm.

It is 4 cm wide and 5 cm high.

Calculate the volume of the cube.

Question 3

A cuboid is 9 cm long, 4 cm wide, and 5 cm high.

Calculate the volume of the cube.

Cylinder

The cylinder is a three-dimensional figure composed of two identical parallel circles called bases, between which the lateral area expands.

Other properties:

The distance between the two bases is constant and is called the height of the cylinder - we will mark it with an $H$
The radius of both bases is equal, we will mark it with an $R$

Volume of the cylinder

The volume contained within the cylinder is usually denoted by $V$ .
Formula to calculate the volume of the cylinder:
$π\times \R^2\times \H = V$

When:

$π$ = PI ( $3.14$ )
$R$ = Radius of the base
$H$ = Height of the cylinder

For more information about the volume of the cylinder click here

Check your understanding

Question 1

An unfolded cuboid is shown below.

What is the surface area of the cuboid?

Question 2

What are the dimensions of a cuboid composed of two 4X3 rectangles

and of four 4X4 squares?

Question 3

Which dimensions may represent a cuboid?

Total surface area of a cylinder

The total area of the lateral surface and the two bases - we will denote it with $A$
We will use the formula:

$2πR\times \H+2π\times \R^2$

When:

$π$ = PI ( $3.14$ )
$R$ = Radius of the base
$H$ = Height of the cylinder

For more information click here

Lateral area of a cylinder

Only the lateral area without the bases. We will denote it with $S$
We will use the formula:

2πR\times \H

When:

$π$ = PI ( $3.14$ )
$R$ = Radius of the base
$H$ = Height of the cylinder

For more information about the cylinder click here

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:

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.

Question 3

Shown 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.

Prism

The right triangular prism is a three-dimensional figure that is composed of 2 triangles and 3 rectangles:

Base of the prism: the 2 triangles that compose it will always be identical (marked in orange).
The triangles can be isosceles, scalene, or equilateral.
To delve deeper into the topic of the bases of the prism click here
Faces of the prism: the 3 rectangles that make up the lateral faces - they will not necessarily be identical.
Heights of the prism: the three lines that join the bases - always have the same length.
For more information about the heights of the prism click here

Let's practice!
In a right triangular prism, are the triangular bases always identical?
Solution:
Yes! The triangles, which are actually the bases, are always the same.
Exercise:
How many heights are there in a right triangular prism?
Are they identical?
Solution:
There are 3 heights in a right triangular prism and they always have the same length.
Exercise:
Do the three rectangles that make up the lateral faces of the prism have to be identical?
Solution:
No.
The edges of the triangle do not necessarily have to be equal and this could create different rectangles.

Volume of the right triangular prism

The volume of the prism is usually expressed through the following formula:
$V= S \cdot H$

$S$ = Area of the base
$H$ = Height of the prism

You can read again about the area of the prism by clicking here

Test your knowledge

Question 1

Look at the the cuboid below.

What is its surface area?

Question 2

Given the cuboid of the figure:

What is its volume?

Question 3

A cuboid is shown below:

What is the surface area of the cuboid?

Area of a right triangular prism

The area of a right triangular prism is, in fact, the total sum of the surfaces of its two bases (the triangles) and its three lateral faces (the rectangles).

For more information about the area of the prism click here

Examples and exercises with solutions of three-dimensional figures

Exercise #1

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

Exercise #2

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

(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 #3

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 #4

Look at the cuboid below:

What is the volume of the cuboid?

Video Solution

Step-by-Step Solution

To determine the volume of a cuboid, we apply the formula:

Step 1: Identify the dimensions of the cuboid:
- Length ( $l$ ) = 12 cm
- Width ( $w$ ) = 8 cm
- Height ( $h$ ) = 5 cm
Step 2: Apply the volume formula for a cuboid:

The formula to find the volume ( $V$ ) of a cuboid is:

$V = l \times w \times h$

Step 3: Substitute the given dimensions into the formula and calculate: $V = 12 \times 8 \times 5$

Step 4: Perform the multiplication in stages for clarity:

First, calculate $12 \times 8 = 96$

Then multiply the result by 5: $96 \times 5 = 480$

Therefore, the volume of the cuboid is $\mathbf{480 \, \text{cm}^3}$ .

Answer

480 cm³

Exercise #5

An orthohedron has the dimensions: 4, 7, 10.

How many rectangles is it formed of and what are their dimensions?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the dimensions of each pair of rectangles given the orthohedron's dimensions.
Step 2: Each face of a cuboid corresponds to a rectangle, with three distinct pairs of dimensions.
Step 3: Identify all pairs of dimensions and count the pairs that form sets of rectangles.

Now, let's work through each step:
Step 1: The orthohedron's dimensions are given as $4$ , $7$ , and $10$ .
Step 2: A cuboid (orthohedron) has three pairs of opposite rectangular faces:
- Pair 1: Two rectangles with dimensions $4 \times 7$ .
- Pair 2: Two rectangles with dimensions $4 \times 10$ .
- Pair 3: Two rectangles with dimensions $7 \times 10$ .
Step 3: Count each of the pairs to verify the total number of rectangles formed.
We find there are 6 rectangles in total, with the dimensions specified above fulfilling the conditions for each face of the cuboid.

The solution to the problem is that the orthohedron is formed of:
2 Rectangles $4 \times 7$ ,
2 Rectangles $4 \times 10$ ,
2 Rectangles $7 \times 10$ .

These dimensions and quantities match choice #3 in the answer options provided.

Answer

2 Rectangles 4X7

2 Rectangles 4X10

2 Rectangles 7X10

Do you know what the answer is?

Question 1

Look at the cuboid below.

What is its surface area?

Question 2

Look at the cuboid below.

What is the surface area of the cuboid?

Question 3

Look at the cuboid below:

What is the volume of the cuboid?

Start practice

Three-Dimensional Figures

Three-dimensional figures

What are three-dimensional figures?

What differences do three-dimensional figures have?

Test yourself on cuboids!

Cuboid

Each cuboid has:

Face diagonals:

Diagonal of the cuboid:

Volume of the cuboid

Surface area of a cuboid

Lateral area of a cuboid

Cylinder

Volume of the cylinder

Total surface area of a cylinder

Lateral area of a cylinder

Prism

Volume of the right triangular prism

Area of a right triangular prism

Examples and exercises with solutions of three-dimensional figures

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Cuboids