Where does all mathematics come from? What shapes are the basis of many calculations and solutions? A cylinder is one of those shapes that takes its place alongside spheres and cubes. But what is a cylinder? How can it be described? What distinguishes one type of cylinder from another? Let this article answer these questions and examine all the most interesting aspects of this three-dimensional shape. Are you ready to learn about the basic properties and formulas behind this mysterious object? Let’s reveal all the secrets of the cylinder together!
What Is a Cylinder: An Overview of Definitions and Types
A cylinder is a three-dimensional solid figure consisting of two identical and parallel circular bases connected by a curved surface. The axis of the cylinder is a straight line passing through the centers of the circular bases and perpendicular to them. And the height of the cylinder is determined by the length of its axis. For example, in the figure below, OO1 indicates the axis of symmetry and the height of the cylinder.
It is clear that in order to properly understand a cylinder, it is worth familiarizing yourself with its radius – it is simply the distance from the center of the base to its outer border, which helps to better imagine the shape of this figure. Returning to our image, we can establish that AO is the radius of the cylinder.
Different types of cylinders
Now that we have figured out what is a cylinder, it is time to consider its different types. Did you know that there are four basic types of cylinders in geometry? Let’s take a look at each of them:
- Right Cylinder: A cylinder whose axis is at an angle of 90 degrees or perpendicular to the base is called a regular cylinder. If the bases are circular, it is called a regular circular cylinder;
- Oblique cylinder: This cylinder has no perpendicular axis, so its sides are inclined above the base and the two bases are not directly in front of each other. It is an inclined shape of a round cylinder. The famous Leaning Tower of Pisa is just such an oblique cylinder;
- Elliptical cylinder: A cylinder whose bases are in the shape of an ellipse is called an elliptical cylinder. It can also be described as a cylinder whose cross-section is ellipse;
- Hollow cylinder: It is a cylinder that consists of two right circular cylinders confined within each other. The base of such a cylinder is similar to a ring, so the bottom of a hollow cylinder resembles an area bounded by two concentric circles.
So, a cylinder is not just a geometric figure, but a whole world of different shapes and types. Understanding them helps us better understand its role in mathematical calculations.
Basic Properties of a Cylinder: What You Need to Know
Each geometric shape has its own characteristics or some properties different from other shapes. And the cylinder is no exception! Let’s look at some key properties of this 3D shape:
- A cylinder has one curved surface and two identical flat faces;
- Two circular bases of the cylinder are congruent to each other;
- The size of the cylinder depends on the radius of the base and the height of the curved surface;
- Unlike a cone, cube, or rectangular parallelepiped, a cylinder has no vertices. This means that there is no specific angle in the cylinder;
- The base and upper part of the cylinder are identical, that is, it has the same base – circular or elliptical.
From Area to Volume: Basic Cylinder Formulas
What is a cylinder without its formulas? How can we calculate its surface area and volume? Let’s look at the three main formulas associated with this interesting geometric object:
| Term | Definition | Formula |
|---|---|---|
| Curved surface area | The curved surface area of a cylinder is the area between its two bases. |  |
| Total surface area of the cylinder | The total surface area of the cylinder is the sum of the areas of the lateral surface and the two circular bases. |  |
| Volume of a cylinder | The volume of a cylinder is the space occupied by the cylinder. |  |
Note: In all these formulas, R is the radius of the base and h is the height of the cylinder. Thanks to these formulas, we can easily calculate various parameters of the cylinder and use them in various mathematical problems.
What is a Cylinder: Practical Examples with Answers and Solutions
Knowing all the formulas, definitions and properties of the cylinder, let’s move on to practice and consider concrete examples for an even deeper understanding of these concepts.
Example 1: What is a cylinder?
A cylinder is a three-dimensional shape that consists of two circular bases connected by a curved surface formed by folding a rectangle. The upper and lower faces of the cylinder are equal.
Example 2: How many vertices does a cylinder have?
A cylinder has one curved surface, two identical flat faces and zero vertices.
Example 3: Write a three main cylinder formulas.
Three main formulas for a cylinder:
- Сurved surface area: CSA=2⋅π⋅R⋅h square units;
- Total surface area: TSA=2⋅π⋅R⋅(h+R) square units;
- Volume: V=π⋅R2⋅h cubic units.
Example 4: A cylinder has a radius of 6 cm and a height of 7 cm. What is its surface area?
So, by the condition of the problem, we have that the lengths R=6 and h=7. This means that we can use these numbers in the formula for the surface area: TSA=2⋅π⋅R⋅(h+R). Let’s put them together and find the answer.
Thus, the surface area of the cylinder is 489.84 cm2.
Example 5: What is the volume of a cylinder with a radius of 4 cm and a height of 5 cm?
Again, we have a radius of R=4 and a height of h=5. It is known that the volume of a cylinder is calculated by the formula V=π⋅R2⋅h. Let’s substitute our numbers and get the answer.
Therefore, the volume of the cylinder is 251.2 cm3.
Explore Deeper: Other Aspects of Cylinder Geometry!
Do you want to understand the formulas and properties of the cylinder in more detail? Here are a few topics you might be interested in:
- Diameter of a cylinder: Formulas and examples – Did you know how to calculate the diameter of a cylinder to better understand its dimensions? Let’s look at it together and solve the examples.
- Total surface area of a cylinder: Formula and examples – How to calculate the curved surface area of a cylinder to determine its volume? Let’s look into this topic, consider the formulas and solve a couple of examples.
- Volume of a cylinder: Formulas and examples – What is behind the calculation of the volume of a cylinder and how can it be applied in practice? Let’s look at the formulas and problem solutions to understand it better.