The cylinder is one of the basic 3D figures, which has two parallel circular bases which are located at a certain distance from each other. The two circular bases are joined by a curved surface. The line segment joining the two centers of the circles is the axis of the cylinder. The distance between the two circular bases is equal to the height of the cylinder. Since the cylinder is a 3D figure, it has two most important basic properties, volume and surface area.
Here, we will learn about some of the most important characteristics of cylinders. In addition, we will learn about the formulas for volume and surface area and use them to solve some exercises.
Definition of a cylinder
A cylinder is a three-dimensional solid that contains two parallel bases connected by a curved surface. The bases are usually circular in shape. The perpendicular distance between the bases is denoted as the height “h” of the cylinder and “r” is the radius of the cylinder.
Fundamental characteristics of a cylinder
The following are some of the most important characteristics of cylinders:
- The cylinder bases are always congruent and parallel to each other.
- If the axis of the cylinder is at a right angle with respect to the base, then it is called a “right cylinder.”
- If one of the bases has an inclination and the axis does not produce a right angle at the bases, then it is called an “oblique cylinder”.
- If the bases are circular, then it is called a “circular cylinder.”
- The bases of a cylinder can also be ellipses. If the base of the cylinder is elliptical, it is called an “elliptical cylinder.”
- A cylinder is similar to a prism in that it has the same cross section throughout.
Important cylinder formulas
Cylinders are three-dimensional figures, so they have two important properties: volume and surface area.
Formula for the volume of a cylinder
We can calculate the volume of a cylinder by multiplying the area of the bases by the height. In the case of circular cylinders, the area of the bases is given by πr². Therefore, the volume of a cylinder is:
| $latex V=\pi {{r}^2}\times h$ |
where r represents the length of the radius of the circular bases and h is the height of the cylinder.
Formula for the surface area of a cylinder
The surface area is equal to the entire surface occupied by the cylinder. We can calculate the surface area by adding the area of the two circular faces and the area of the curved lateral surface. Each circle has an area of πr² and the lateral surface has an area of 2πrh, so the total surface area is:
| $latex A_{s}=2\pi {{r}^2}+2\pi r h$ |
Examples of cylinder problems
The following exercises are solved by applying the cylinder formulas seen above.
EXAMPLE 1
What is the volume of a cylinder that has a radius of 4 m and a height of 5 m?
Solution: We have the lengths $latex r=4$ and $latex h=5$. Therefore, we use these values in the volume formula:
$latex V=\pi {{r}^2}\times h$
$latex V=\pi {{(4)}^2}\times 5$
$latex V=\pi (16)\times 5$
$latex V=251.3$
The volume of the cylinder is 251.3 m³.
EXAMPLE 2
If a cylinder has a radius of 7 m and a height of 10 m, what is its volume?
Solution: We use the volume formula with the given values:
$latex V=\pi {{r}^2}\times h$
$latex V=\pi {{(7)}^2}\times 10$
$latex V=\pi (49)\times 10$
$latex V=1539.4$
The volume of the cylinder is 1539.4 m³.
EXAMPLE 3
A cylinder has a radius of 6 m and a height of 7 m. What is its surface area?
Solution: We have the lengths $latex r=6$ and $latex h=7$, so we use these values in the formula for surface area:
$latex A_{s}=2\pi {{r}^2}+2\pi r h$
$latex A_{s}=2\pi {{(6)}^2}+2\pi (6)(7)$
$latex A_{s}=226.2+263.9$
$latex A_{s}=490.1$
The surface area is 490.1 m².
EXAMPLE 4
What is the surface area of a cylinder that has a radius of 8 m and a height of 9 m?
Solution: We can use the values given in the formula for surface area:
$latex A_{s}=2\pi {{r}^2}+2\pi r h$
$latex A_{s}=2\pi {{(8)}^2}+2\pi (8)(9)$
$latex A_{s}=402.1+452.4$
$latex A_{s}=854.5$
The surface area is 854.5 m².
See also
Interested in learning more about cylinders? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
