What are Cylinders?
A ‘cylinder’ is a three-dimensional object that is made up of a curved surface with a circular top and base. This special structure of cylinders makes them stand perfectly upright if they are placed on one of their two bases. Cylinders can also be made to roll if they are on their side. The objects we see everyday like pipes, fire extinguishers, water tanks, pencils, and so on, are some of the real-life examples of cylinders.
What are the Properties of Cylinders?
Just like every geometrical shape, cylinders do have a few properties. Let’s look at what those are:
- A cylinder has two identical circular faces and one curved lateral surface.
- The volume and area of a cylinder is determined by the base radius and the height, which is the perpendicular distance between the two bases.
- Unlike a cone, a cube, or a cuboid, a cylinder does not have a vertex. It shows there is no specific corner in any cylinder.
Formulas of Cylinders
A cylinder has three major formulas related to its surface area and volume:
Lateral Surface Area of a Cylinder
The area of the curved surface of a cylinder is known as the ‘lateral surface area’ or ‘curved surface area’.
The formula for the lateral surface area is stated as follows:
Lateral surface area (LSA) = 2\( \pi \)rh, where h is the height and r is the radius.
Surface Area of a Cylinder
The total area of the cylinder, that is the area of the bases and the lateral surface area is called the ‘surface area’ of the cylinder.
The formula for the surface area of a cylinder is stated as follows:
Surface area = 2\( \pi \)r (h + r), where, h is the height and r is the radius.
Volume of a Cylinder
The volume of a cylinder is the amount of space occupied by the cylinder. For instance, if we have to fill a cylinder with water, then the amount of water that can be filled is the volume of the cylinder.
The formula for the volume of a cylinder is as follows:
Volume of cylinder = \( \pi \)r\( ^2 \)h , where, h is the height and r is the radius.
Rapid Recall
Solved Cylinder Examples
Example 1: What will be the volume of a cylinder that has a radius of 6 units and a height of 10 units?
Solution:
The details provided in the question are, the radius of the cylinder is r = 6 units and height h = 10 units. As we know the volume of the cylinder can be calculated by the formula,
Volume of cylinder = \( \pi r^2h \)
= \( \frac{22}{7}\times (6)^2\times 10 \)
= \( 1131.43 \) cubic units
Hence, the volume of a given cylinder is 1131.43 cubic units.
Example 2: Emily bought a new cylindrical water bottle so she can gift it to her friend Rosalie. The radius of the bottle is 2 cm and the height is 8 cm. She wants to wrap the bottle with a rectangular gift-wrap sheet of length 20 centimeters and width 10 centimeters. Check whether the bottle could be covered using the gift-wrap sheet or not.
Solution:
Given, the radius of the water bottle r = 2 cm and the height of the bottle h = 8 cm.
To check whether the water bottle could be wrapped with the gift-wrap sheet, we need to find the area of the gift-wrap sheet and the surface area of the water bottle, and compare the areas of these two objects.
As we know the bottle is in the shape of a cylinder.
So, surface area = 2\( \pi \)r (h +r)
= 2 \( \times \) 3.14 \( \times \) 2 \( \times \) (8 + 2)
= 2 \( \times \) 3.14 \( \times \) 2 \( \times \) (10)
= 125.6 square centimeters
The gift-wrap sheet is in the shape of a rectangle.
Therefore, the area of the gift-wrap sheet = length \( \times \) width
= 20 \( \times \) 10
= 200 square centimeters
The area of the gift-wrap sheet is greater than the surface area of the cylinder. Therefore, the water bottle can be wrapped using the gift-wrap sheet.
Example 3: Ross is doodling in his room using different colored pencils that have a height of 17 cm and radius of 0.4 cm. He then realized that these color pencils are also cylinders. Find the lateral surface area of the pencils.
Solution:
As provided in the question, the height of the pencils is 17 cm and the radius is 0.4 cm. It is also given that the shape of the pencils are cylinders. So,
Lateral Surface Area = 2\( \pi \)rh
= 2 \( \times \) 3.14 \( \times \) 0.4 \( \times \) 17
= 42.704 square centimeters
Therefore, the lateral surface area of the colored pencils is 42.704 cm.
A cylinder has two circular bases joined together with a curved lateral surface. So a cylinder is a three-dimensional object.
A cylinder has two circular faces and one curved surface.
The point at which the sides or edges of a shape meet is known as the vertex. There is no such point of intersection with reference to a cylinder. Hence, a cylinder does not have any vertices.
A slanted cylinder is an oblique cylinder. In other words, when the bases of the cylinder are not exactly aligned, then, the cylinder is termed as an oblique cylinder.