Surface Area of Cylinder (CSA and TSA) |Formula, Derivation, Examples
Surface Area of a Cylinder is the amount of space covered by the flat surface of the cylinder’s bases and the curved surface of the cylinder. The total surface area of the cylinder includes the area of the cylinder’s two circular bases as well as the area of the curving surface.
Let’s learn about the formulas for Curved Surface Area (CSA) and Total Surface Area of Cylinder (TSA)and how to use them with the help of solved examples.
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What is Surface Area of Cylinder?
Surface area of a cylinder is the total area that covers its outer surface.
Let us magine a cylindrical object, like a can or a pipe. To find its surface area, we need to consider two parts:
- Curved Surface Area (CSA): This is the area of the curved side of the cylinder. You can think of it as if you were peeling off the label of a can. It’s like the wrapper around the cylinder.
- Two Circular Ends: A cylinder has two circular ends, one at the top and one at the bottom. Each of these circular ends has an area of Ï€r^{2}.
Surface Area of Cylinder Formula
Because a cylinder has a curved surface, we can express both its curved surface area and overall surface area.
Here are the formulas for the two types of surface areas of cylinder, with radius = r and height = h.
Formula | Value |
---|---|
Curved Surface Area of Cylinder | 2Ï€rh |
Total Surface Area of Cylinder | 2Ï€r^{2} + 2Ï€rh = 2Ï€r(r + h) |
Now, lets learn about both of them in detail.
Curved Surface Area (CSA)of Cylinder
The curved surface area of the cylinder is enclosed between the two parallel circular bases. It is also known as the lateral surface area.
CSA of Cylinder Formula
The Curved Surface Area (CSA) of Cylinder formula is as follows:
Curved Surface Area = 2Ï€rh sq. units
where,
- r is the Radius of Cylinder
- h is the Height of Cylinder
Total Surface Area (TSA) of Cylinder
A cylinder’s total surface area is the sum of its curved surface area and the area of its two circular bases. It is calculated by summing the areas of the two bases and the curved surface (CSA).
TSA of Cylinder Formula
The formula for Total Surface Area (TSA) of cylinder is given by,
Total Surface Area of Cylinder Â = 2Ï€r^{2} + 2Ï€rh = 2Ï€r(r + h) sq. units
where,
- r is Radius of Cylinder
- h is Height of Cylinder
Surface Area of Cylinder Derivation
Let us consider a cylinder whose radius is r and height is h. The cylinder is divided into three parts: one circular base at top, one rectangular curved area and another circular base at bottom.
- The rectangular area has length of 2Ï€r and breadth of h. So, the area is, A_{1} = 2Ï€rh, which is also the curved surface area of the cylinder.
Hence , the formula for the CSA of cylinder is given by
CSA of cylinder = 2Ï€rh
- The area of a circular base with radius r = Ï€r^{2}. So, the area of two such bases is, A_{2} = (Ï€r^{2} + Ï€r^{2}) = 2Ï€r^{2}.
Now, the total surface area of the cylinder is the sum of above two areas.
A = A_{1} + A_{2}
AÂ = 2Ï€r^{2} + 2Ï€rh
TSA of cylinder = 2Ï€r(r + h)
This derives the formula for total surface area of a cylinder.
Difference between TSA and CSA of Cylinder
The main differences between them Total Surface Area and Curved Surface Area are tabulated below.
Property | Total Surface Area (TSA) of Cylinder | Curved Surface Area (CSA) of Cylinder |
---|---|---|
TSA vs. CSA of Cylinder | ||
Definition | The total area of all the surface which includes the curved surface and the two base areas. | It is defined as the area of the curved surface of the cylinder. |
Formula | The formula for TSA of the cylinder is, TSA = Â 2Ï€r (r + h) | The formula for CSA of the cylinder is, CSA = 2Ï€rh |
Relation | TSA is greater than CSA as it includes CSA along with both the base areas. | CSA is lesser than TSA. |
How to Calculate Surface Area of Cylinder?
Surface area of a cylinder can be calculated using the steps added below,
Step 1: Note the radius, ‘r’, and height, ‘h’ of cylinder. Remember both have the same units. Here, given r = 14 cm, h = 10 cm
Step 2: Find the total surface area of the cylinder, the formula for the total surface area of the cylinder = 2Ï€r(r + h)
Step 3: Put the given values in the above formulas and find the answer in square units.
Surface Area of Cylinder in square meters
Let’s find the total surface area of a cylinder that has a radius of 14 cm and a height of 10 cm.
Substitute the values in the formula we get,
Total Surface Area(TSA) = 2Ï€r(r + h)
TSA = 2Ï€ Ã— 14(14 + 10)Â
TSA = 2Ï€ Ã— 336Â
TSA = 2 Ã— 3.14 Ã— 336
TSA = 2110.08 square cm
Surface Area of Cylinder in square feet
Let’s calculate the total surface area of a water tank with a radius of 4 feet and a height of 8 feet in square feet.
Substitute the values into the formula:
TSA = 2Ï€ Ã— 4 Ã— (4 + 8)
Now, let’s calculate the values inside the brackets.
TSA = 2Ï€ Ã— 4 Ã— 12 = 96Ï€ square feet â‰ˆ 96 Ã— 3.14 square feet
â‰ˆ 301.44 square feet (rounded to two decimal places)
Volume of Cylinder
Volume of a cylinder is defined as the total amount of space occupied by the cylinder. For a cylinder of base radius r, and height h the volume is given by the formula,
Volume of Cylinder = Ï€r^{2}h
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Surface Area of Cylinder Examples
Let’s solve some questions on the formulas of TSA and CSA of a cylinder.
Example 1: Find the curved surface area of the cylinder of radius 3 cm and height of 7 cm.
Solution:
Given,
- r = 3
- h = 7
Curved Surface Area of Cylinder(CSA) = 2Ï€rh
CSA = 2 (22/7) (3) (7)
CSA = 2 (22) (3)
CSA = 132 cm^{2}
Example 2: Find the radius of the cylinder of curved surface area of 220 sq. cm and height of 7 cm.
Solution:
Given,
- A = 220
- h = 7
Curved Surface Area of Cylinder(CSA) = 2Ï€rh
220 = 2 (22/7) (r) (7)
220 = 44r
r = 220/44
r = 5 cm
Example 3: Find the total surface area of the cylinder of radius 21 cm and height of 42 cm.
Solution:
Given,
- r = 21
- h = 42
Total Surface Area(TSA) = 2Ï€r^{2} + 2Ï€rh
TSA = 2 (22/7) (21) (21) + 2 (22/7) (21) (42)
TSA = 2 (22) (3) (21) + 2 (22) (3) (42)
TSA = 2772 + 5544
TSA = 8316 sq. cm
Example 4: Find the total surface of the cylinder if the curved surface area is 176 sq. cm and the height is 21 cm.
Solution:
Given,
- A = 176 cm^{2}
- h = 21 cm
Curved Surface Area of Cylinder(CSA) = 2Ï€rh
176 = 2 (22/7) (r) (21)
176 = 2 (22) (r) (3)
r = 176/132
r = 1.33 cm
Total Surface Area(TSA) = 2Ï€r^{2} + 2Ï€rh
TSA = 2 (3.14) (1.33) (1.33) + 176
TSA = 11.10 + 176
TSA = 187.1 sq. cm
Cylinder Surface Area Practice Questions
Here is a worksheet on Surface Area of Cylinder for you to solve.
Q1. If the Radius of a cylinder is 5 cm and the height of the cylinder is 15 cm. Find the curved area of the cylinder.
Q2. If the Radius of a cylinder is 12 m and the height of the cylinder is 21 m. Find the total area of the cylinder.
Q3. What is the radius of a cylinder with height of the cylinder is 21 cm and curved surface area 225 cm^{2}?
Q4. What is the height of a cylinder with radius of the cylinder is 21 cm and curved surface area 105 cm^{2}?
Cylinder Area Formula FAQs
What is a Cylinder?
A cylinder is a three-dimensional shape having two circular bases in parallel to each other joined by a curved surface.
How to Find the Surface Area of Cylinder?
For finding the surface area of a cylinder, we will find the surface area of curved surface and area of the circular bases of the cylinder. Now add all the areas to get the total surface area.
What is the TSA of Cylinder?
For the cylinder of radius “r” and height “h” the TSA (total surface area )of cylinder formula is,
- Total Surface Area(TSA) = 2Ï€r (h + r) sq. unit
What is CSA of Cylinder?
The CSA (Curved Surface Area)of Cylinder is given by the following formula
Curved Surface Area(CSA) = 2Ï€rh sq. unit
What is the Formula of Volume of Cylinder?
For the cylinder of radius “r” and height “h” the formula for finding the volume of a cylinder is,
Volume of cylinder (V) = Ï€r^{2}h cubic units
What is Surface Area of Cylinder with one side open?
Surface area of a cylinder with one side open can be calculated by finding the area of bottom circular base and the curved surface of the cylinder and then adding both the result. Thus,
Surface Area of an Open-Top Cylinder = Ï€r(r + 2h)
What is Surface Area of Hollow Cylinder?
For a hollow cylinder with an outer radius R and inner radius r the inner surface area is defined as the curved area of the inner surface of the cylinder. It can be calculated using the formula,
Inner Surface Area = 2Ï€rh