Surface Area of Cylinder - Introduction
A cylinder is a three-dimensional solid shape having two parallel circular bases joined by a curved surface at a fixed distance. It can also be seen as a set of circular discs stacked on one another.
The line segment joining the centers of circular bases is called the axis of the cylinder. The distance between the axis to the outer curved surface is called radius (r) of the cylinder and the perpendicular distance between the two parallel circular bases is called height (h) of the cylinder.
Examples of cylinders: candles, water tank, pipes, well, etc.
The subject experts at Vedantu have compiled the article to help the students learn the concepts with ease. Follow the full article to learn about the following concepts -
Surface Area of Cylinder
In geometry, the area of any shape is the region covered by it in a plane. A cylinder consists of two types of surface, one is a curved surface and the other is a circular basis. The area of both circular bases is equal.
The Surface Area of the Cylinder can be Classified into Two Types:
Curved surface area (CSA)
Total surface area (TSA)
Curved Surface Area (CSA) of Cylinder
The curved surface area of a cylinder is defined as the area of its curved surface or the area obtained after excluding the area of its two circular bases. It is also called Lateral surface area (LSA).
The CSA of a cylinder having its base radius ‘r’ and height ‘h’ is given by:
Curved surface area (CSA) of cylinder = 2πrh sq. units.
Total Surface Area (TSA) of Cylinder
The total surface area of a cylinder is defined as the sum of areas of its curved surface and two circular bases.
The TSA of a cylinder having its base radius ‘r’ and height ‘h’ is given by:
Total surface area (TSA) of cylinder = Area of curved surface + Area of its two circular bases
or,
TSA of cylinder = CSA of cylinder + Area of its two circular bases
= 2πrh + 2πr2 sq. units
So, TSA of cylinder = 2πr (h + r) sq. units
Derivation of the Formula of Surface Area of Cylinder
Consider a situation in which a cylinder has to be covered with colored papers. The covering has to be done with a minimum amount of paper. So, to cover the cylinder first take a rectangular sheet of paper, whose length (l) is just enough to go round the cylinder and the breadth is equal to the height (h) of the cylinder as shown in the figure below.
(Images will be updated soon)
It is to be noted that the length (l) of the rectangular sheet is equal to the circumference of the circular base which is equal to 2πr.
The area of the rectangular sheet gives the curved surface area of the cylinder.
So, the curved surface area of the cylinder = Area of the rectangular sheet of paper
= length × breadth
= Circumference of the circular base × h
= 2πr × h
Therefore, the curved surface area of the cylinder = 2πrh sq. units
Where r is the radius of the base and is the height of the cylinder.
Now, if the top and the bottom of the cylinder are also to be covered with colored papers, then we need two circular regions to do that, each of radius r and having an area of πr2 each.
So, total surface area of cylinder = Area of curved surface + Area of two circular regions
= CSA of cylinder + πr2 + πr2
= 2πrh + 2πr2
Therefore, Total surface area of cylinder = 2πr (h + r) sq. units
Where r is the radius of the base and h is the height of the cylinder.
Solved Examples:
Q.1. A cylindrical pillar is 50 cm in diameter and 7 m in height. Find the cost of painting the curved surface of the pillar at the rate of ₹12 per sq. mts.
Solution: Given, height(h) of cylindrical pillar = 7 m
And, radius (r) of circular base =50/2 cm = 25 cm = 0.25 m
So, the curved surface area of the cylindrical pillar = 2πrh sq. units
= 2 ×(22/7)× 0.25 × 7
= 2 ×(22/7)× (25/100) × 7
= 11 sq. mts
Given, the cost of painting 1 sq. mts of area is ₹12.
∴ the cost of painting 11 sq. mts of area = 11 × 12 = ₹132
Therefore, the cost of painting the curved surface of the pillar at the rate of ₹12 per sq. mts. is ₹132.
Q.2. The curved surface area of a right circular cylinder of base radius 7 cm is 110 cm2. Find the height of the cylinder.
Solution: Given, the base radius (r) of the cylinder = 7 cm.
And, curved surface area of cylinder = 110 cm2
Let the height of the cylinder be h.
Then, 2πrh = 110 cm2
⇒ 2 ×(22/7)× 7 × h = 110 cm2
⇒ h = 2.5 cm.
Therefore, the height of right circular cylinder is 2.5 cm
Q.3. How many square meters of metal sheet is required to make a closed cylindrical tank of height 1.8 m and base diameter 140 cm?
Solution: Given, height of closed cylindrical tank = 1.8 m
And, the radius (r) of circular base = 140cm/2 = 70 cm = 0.7 m
So, area of metal sheet required to make closed cylindrical tank = total surface area of cylinder = 2πr (h + r)
= 2 ×(22/7)× 0.7 × (1.8 + 0.7)
= 2 ×(22/7)× 0.7 × 2.5
= 2 ×(22/7)×(7/10)×(25/10)
= 11 sq. mts
Therefore, the area of the metal sheet required to make a closed cylindrical tank is 11 sq. mts.
FAQs on Surface Area of Cylinder
1. How to find the volume of a cylinder?
The volume of any object is the capacity of that object. So the volume of objects differs according to their size and shapes. So, to find the volume of a cylinder we need to know the height of the cylinder and the surface of the base. That is the volume is equal to πr2h where r is the radius of the base of the cylinder and h is the height or the depth you can say. Follow the full article on how to find the volumes of the different objects on Vedantu's website or mobile app.
2. How to find out the surface area of a hollow cylinder?
In this case you need to keep in mind a few things that are the radius of the outer circle represented by R, the radius of the inner circle as the hollow part of the cylinders makes an inner circle which is represented by r, and finally the height of the cylinder which is represented as h. So, the external curved surface area is 2πRh (small r is replaced by capital R). And, the internal curved surface area is 2πrh (small r for inner circle)
3. How to find the area of the ring between the outer circle and an inner circle in a hollow cylinder?
Follow the other discussion where we have explained the process to find out the surface area of a hollow cylinder. So to find the area of the ring we simply need to subtract the area of the inner circle from the area of the outer circle. Hence, the area of the circular end is π(R2 - r2).
4. What is the difference between the curved surface area and lateral surface area of a cylinder?
Lateral surface area is that area that covers all the area of a solid object except its base and the top area. Imagine the four sides of a wall leaving the roof and the floor of the room. So the area of a cylinder excluding its top and bottom area will be its LSA. Whereas curved itself means something which is folded, a cylinder is a curved object so ideally, we do not use LSA in cylinder or cone. Here we make use of CSA to find out the area. Though the process is the same we exclude the top and the bottom areas in the calculation.
5. How important is the topic of the surface area of a cylinder?
The concept is one of the most important portions of geometry. It has various real-life applications like designing, wrapping an object, determining the quantity to cover a cylindrical shape object, painting that object, etc. Learning should be practical and it should be fun, it is the sole motto we believe in and guide our students in the same direction.
