# 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 $cm^3$ .

## 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

#### 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\)

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### 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

## 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

### 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

### 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

## 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.

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

### 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).