What is a Cylinder?
A cylinder is a three dimensional shape formed by two parallel and congruent circular bases joining a curved surface between them. The distance between the centers of the two circular bases is fixed and is known as the height of the cylinder. The radius of the base is referred to as the radius of the cylinder.
Some everyday examples of cylindrical shape are soft drink cans, beakers, pipes, batteries and so on.
Lateral Surface Area of a Cylinder
The curved surface of a cylinder is also known as its lateral surface. The area of this lateral surface is referred to as the lateral surface area of the cylinder.
We can calculate the lateral surface area of a cylinder by multiplying the circumference of its circular base by its height.
Lateral Surface Area of Cylinder, LSA = Base Circumference x height
Lateral Surface Area of Cylinder, LSA = 2πr x h
Lateral Surface Area of Cylinder, LSA = 2πrh
We know that the radius ‘r’ is half of the diameter ‘d’, so the formula can also be written as:
Lateral Surface Area of Cylinder, LSA = πdh
where,
‘d’ denotes the diameter of the cylinder
‘h’ denotes the height of the cylinder
‘π’ is a math constant that is equal to \(\frac{22}{7}\) or 3.14.
Like the surface area of different geometric shapes, lateral surface area is also measured in square units.
[Note: We can also calculate the lateral surface area of a cylinder by subtracting the total base area from the total surface area of the cylinder.]
Fun Fact
If you cut open the curved surface of a cylinder you will notice that it is a rectangle with length 2πr and width h.
Solved Examples
Example 1:
A model of a cylindrical cement tanker has a radius of 100 centimeters. Find the lateral surface area of the tanker if its height is 200 centimeters? [Take π = 3.14]
Solution:
As stated in the question:
Radius, r of the cylindrical tanker = 100 cm
Height, h of the cylindrical tanker = 200 cm
Apply the formula for the lateral surface area of the cylinder:
Lateral Surface Area of Cylinder, LSA = 2πrh
= 2 x 3.14 x 100 x 200 [Substitute values]
= 125600 cm\(^2\) [Multiply]
Therefore, the lateral surface area of the cylindrical tanker is 125600 square centimeters.
Example 2: The total surface area of a cylinder is 200 square inches. Find the lateral surface area of the cylinder if the area of its circular base is 50 square inches?
Solution:
As stated in the question:
Total surface area of cylinder = 200 square inches
Base surface area of cylinder = 50 square inches
We know that a cylinder has two circular bases, so its total base area will be:
2 × 50 = 100 square inches
Lateral Surface Area = Total surface area – Total Base Area
= 200 – 100 [Substitute values]
= 100 inch\(^2\) [Subtract]
Therefore, the lateral surface area of the cylinder is 100 square inches.
Example 3: A cylindrical water barrel is 1.2 meters in height. What is the radius of the barrel if its lateral surface area is 50 square meters? [Take π = 3.14]
Solution:
As stated in the question:
Height, h of the cylindrical barrel = 1.2 meters
Lateral surface area of the cylindrical barrel = 50 square meters
Apply the formula for the lateral surface area of the cylinder:
Lateral Surface Area of Cylinder, LSA = 2πrh
50 = 2 x 3.14 x r x 1.2 [Substitute values]
50 = 7.536 x r [Multiply]
\(\frac{50}{7.536}\) = r [Divide each side by 7.536]
6.63 = r [Simplify]
⇒ r = 6.63 m
Therefore, the radius of the cylindrical water barrel is 6.63 meters.
The measure of the area of the surface of any solid shape, excluding its top and base, is referred to as its lateral surface area.
The total surface area of a cylinder is the sum of the area of its two circular bases and the lateral surface area.
The lateral surface area can never be zero or negative. This is due to the fact that all 3D objects have surfaces and so all such objects will have a positive surface area.