What is a Cylinder?
Two parallel circular bases joined by a curve surface make up the three-dimensional solid known as a cylinder. The height ‘h’ of the cylinder stands for the perpendicular distance between the bases, while ‘r’ stands for the radius of the cylinder.
Formulas related to a Cylinder
The lateral Surface area of a cylinder = 2πrh square units.
The surface area of a cylinder, S = 2πr(r + h) square units.
The volume of a cylinder, V = πr\(^2\)h cubic units.
Where,
‘r’ is the radius of the cylinder
‘h’ is the height of the cylinder
Properties of a Cylinder
A few of the important properties of a cylinder are as follows:
- The bases of the cylinder are always parallel and congruent to one another.
- A cylinder is referred to as a ‘Right Cylinder’ if its axis is at a right angle to its base and the bases are perfectly above one another.
- A cylinder is referred to as an ‘Oblique Cylinder’ if one of the bases of the cylinder is displaced sideways and the axis does not make a right angle with the base.
- A circular cylinder is created when the locus of a line moves parallel to and at a fixed distance from the axis.
- Since a cylinder has a uniform cross section throughout, it resembles a prism but is not a prism.
Rapid Recall
Solved Examples
Example 1:
The radius of a cylinder is 4 centimeters and its height is 8 centimeters. What is the volume of this cylinder?
Solution:
Radius, r = 4 cm
Height, h = 8 cm
Use formula for volume of cylinder.
V = πr\(^2\)h [Write the formula]
V = \(\frac{22}{7} \times 4^2 \times 8\) [Substitute the values]
V = \(3.14 \times 16 \times 8\) [Simplify]
V = 401.92 cm\(^3\) [Multiply]
Therefore the volume of the cylinder is approximately 402 cubic centimeters.
Example 2:
What is the lateral surface area of a cylinder that has a height of 12 meters and a radius of 5 meters.
Solution:
The height of the cylinder is 12 m.
The radius of the cylinder is 5 m.
The lateral Surface area of a cylinder is calculated by applying the formula,
S = 2πrh [Write the formula]
= 2 x 3.14 x 5 x 12 [Substitute the values]
= 376.8 m\(^2\) [Simplify]
The lateral surface area of the cylinder is 376.8 square meters.
Example 3:
A cylindrical-shaped sculpture needs to be painted. The type of paint used is an expensive variety that gives a rich texture and a glossy finish, and which costs about 0.10 $ per square centimeter. The radius of the cylindrical sculpture is 15 centimeters and the height of the same sculpture is 50 centimeters. Find the cost of painting the outer portion of the entire sculpture.
Solution:
The radius of the cylindrical vessel = 15 cm
The height of the cylindrical vessel = 50 cm
The total surface area of the cylinder is calculated as per the formula,
S = 2πr(r + h) [Write the formula]
= 2 x 3.14 x 15(15 + 50) [Substitute the values]
= 2 x 3.14 x 15 x 65 [Simplify]
= 6123 cm\(^2\)
The cost of painting the outer portion of the sculpture is,
= 6123 x 0.10
= 612.3 $
So, it will cost 612.3 dollars to paint the sculpture.
The lateral surface of a cylinder only covers the surface area of the portion excluding the circular bases of the cylinder. The total surface area considers the area of the entire(curved surface and bases)surface of the cylinder.
The ‘volume of a cylinder’ explains the total space occupied by a cylinder with specific dimensions.
When the axis of a cylinder makes an angle other than a right angle, a cylinder is said to be oblique.