# Cylinder

The cylinder is a three-dimensional shape consisting of two bases (two identical circles, parallel to each other) and a lateral surface located in the middle between these two bases that surrounds and connects them.

To better illustrate the definition, you may be impressed by the following figure:

## Cylinder Characteristics

• The cylinder is bounded by two circles and the intersection connecting them.
• The two circles overlap and are parallel to each other (since they are three-dimensional, this means that the two planes in which the circles are also parallel to each other).
• Each of the circles is called a base.
• The distance between the bases is fixed and is called the height of the cylinder (in the drawings $H$, it is actually also the width of the rectangle in the second drawing).
• The radius of each of the two circles is defined as $R$
• Each cylinder is defined by its radius $(R)$ and height $(H)$.

Before addressing the geometrical characteristics of the cylinder, we will first clarify a number of important concepts related to the shape of the cylinder.

In this figure, you can see a cylinder, while in the drawing on the next line, you can get an impression of the arrangement of that cylinder, if we were to disassemble it.

• The cylinder is bounded by two circles and the lateral intersection connecting them.
• The two circles overlap and are parallel to each other (since they are three-dimensional, this means that the two planes in which the circles are also parallel to each other).
• Each of the circles is called a base.
• The distance between the bases is fixed and is called the height of the cylinder (in the drawings $H$, it is actually also the width of the rectangle in the second drawing).
• The radius of each of the two circles is defined as $R$
• Each cylinder is defined by its radius $(R)$ and height $H$.

## Calculation of Cylinder Volume

The volume of the cylinder is basically the volume trapped between the two bases and the side.

To calculate it we need the radius of the cylinder $(R)$ and the height of the cylinder $(H)$.

In geometry it is customary to identify the volume with the letter $(V)$.

The formula for the volume is:

$V=\pi\times R²\times H$

• π (pi) is a mathematical constant whose value is (approximately) equal to. $3.14$.
Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today

## Calculation of Cylinder Area

When we talk about cylinder area, we must distinguish between total surface area and lateral surface area.

Total surface area is the sum of the areas of the two bases and the lateral area (identified by $A$). The area of the base is $\pi R²$

And to obtain the area of the two bases we multiply by $2$, that is,

$2X\pi R²$

To calculate the lateral area, we return to the sliced form. The width of the rectangle is $H$ while the length of the rectangle (we identify it by $L$ ) is equal to the circumference of the circle. The circumference of the circle is calculated by the formula. $R\times L=2X$.

From here we get that the side area is the area of the sliced rectangle, and we must multiply the length of the rectangle by the width of the rectangle.

The area of the resulting rectangle is $R\times H=2X$.

To obtain the total surface area, we add the area of the two bases and the side area. We will eliminate a common factor outside the parentheses. $R 2X$ We obtain the following formula:

$A=2XR(R+H)$

Lateral surface area is the lateral surface area only, without the bases (identified as S). That is, we refer to the area of the sliced rectangle, which we have already calculated for the total surface area.

The formula is:

$S= 2XRH$

## Cylinder Examples and Practice

### Exercise 1

Given the cylinder shown in the drawing.

According to the data, find the volume of the cylinder, the lateral surface area and the total surface area.

Solution:

From the figure it can be seen that the radius of the bases is equal to $R=5$ cm, and the height of the cylinder is equal to $H=10$ cm.

Now it remains only to replace in the formulas that we have learned.

Calculation of the volume of the cylinder:
$V=\pi\times R²\times H$

$= 3.14\times 25\times 10= 785$

Lateral surface area:

$S= 2\times\pi\times R\times H = 2\times 3.14\times 5\times 10= 314$

Total surface area:

$A= 2\times\pi\times R\times (R+H) = 2\times 3.14\times 5( 5+10)= 471$

Answer: Volume of the base $785\operatorname{cm}^3$, lateral surface area $314$ cm², total surface area $471$ cm².

### Exercise 2

Given the cylinder shown in the drawing.

The area of the base is equal to $12$ cm². According to the data, find the volume of the cylinder.

Solution:

From the figure it can be seen that the height of the cylinder is $H= 7$ cm.

We will use the formula for the volume of the cylinder:

$V=\pi\times R²\times H$

Note that the base area is calculated by the formula. $\pi R²$

And in fact it is part of the volume formula.

The area of the base we know from the datum, it is equal to $12$ cm², so we are left to place and obtain:

$\pi R²=V=\pi\times R²\times H$
$X H = 12\times 7= 84$

Answer: The volume of the cylinder is $84$ cc.

If this article interests you, you may be interested in the following articles:

Cylinder volume

Right triangular prism

Bases of a right triangular prism

The lateral faces of a prism

Lateral edges of a prism

Height of a prism

Surface area of triangular prisms

On Tutorela's blog you will find a wide variety of mathematical articles.