Curved Surface Area of Cylinder – Derivation, Formula and Examples

What is a Cylinder?

A cylinder is a 3D solid object with two parallel circular bases at both ends of the cylindrical shape. The radius of the circular bases of the cylinder corresponds to the radius (r) of the cylinder. The axis connecting the two circular bases denotes the height (h) of the cylinder. 

The Surface Area of a Cylinder

In terms of a cylinder, it is imperative to differentiate between the total surface area and the lateral surface area. In layman terms, people tend to relate surface area to total surface area. Hence, making it essential to mark the difference between the two.

If someone asks you to find the surface area, they are probably asking you to figure out the TSA of the cylinder. But before diving into the derivation of the formula of CSA and TSA of a cylinder, you must be well aware of the radius, diameter, height, and π. All these determinants play a crucial role in finding the surface area of a cylinder.

As defined earlier, the radius (r) emerges from the two circular faces while the height (h) emerges from the cylindrical body. Therefore, the surface area calculation involves the area of the two circles, the height of the cylinder, and π.

Curved Surface Area of a Cylinder

The curved surface area of a cylinder is known as the area of the curved surface that connects the two circular bases. 

You can also obtain the CSA of a cylinder after excluding the circular areas from the total area of the cylinder. 

CSA of a Cylinder formula

Considering the height of the cylinder as ‘h’ and radius of the cylinder as ‘r’, we have the formula of the CSA of the cylinder as 2πrh square units.

Derivation of the Curved Surface Area of a Cylinder 

Consider a solid cylindrical shape of radius ‘r’ and height ‘h’. To obtain the formula of the curved surface of the cylindrical body, take a rectangular sheet and wrap it around this cylinder. Cut the edges from the top and bottom to match the shape of this cylinder. The area of this rectangular piece of paper is the curved surface of the cylinder. 

The area of the curved surface of the cylinder is equal to the area of the rectangular sheet.

Therefore, the area of the curved surface = length of the rectangle x breadth of the rectangle.

From this, we can infer that,

Length of the rectangle = Circumference of the base of the cylinder

                                           = 2πr

Breadth of the rectangle = Height of the cylinder

                                             = h

Area of the curved surface = area of the rectangle 

                                                = length x breadth

                                              = 2πrh

Hence, the Curved Surface Area of a Cylinder = 2πrh

Total Surface Area of a Cylinder

Finding the total surface area of a cylinder includes calculating the sum of areas of all surfaces. As mentioned earlier, a cylinder has two types of surfaces, a curved and a circular; the sum of these two surface areas gives us the TSA of a cylinder. 

Total surface area of a cylinder formula

The total surface area of a cylinder is the sum of its lateral curved surface area and the area of two circular bases.

TSA = CSA + area of circular bases

Considering the height of a cylinder to be ‘h’ and radius as ‘r’, the total surface area will be:

\text { TSA }=2 \pi r h+2 \pi r^{2}

Hence, the Total Surface Area of a Cylinder = 2πr(h + r)

Solved Examples for calculating the Surface Area of a Cylinder

Question 1: Ishan’s parents are planning to renovate his room, and Ishan has demanded to decorate his play area with beautifully painted cylindrical tins. Calculate the cost of painting one cylindrical tin if the radius and height of the cylinder are 7 m and 15 m, respectively. The cost of painting the cylindrical containers is ₹ 2 / m^{2} .
(Take π=\frac{22}{7})

Solution:

Radius of cylindrical tin (r) = 7 m

Height of cylindrical tin (h) = 15 m

To find the area of painting, we need to find the total surface area of the cylinder.

Total surface area

=2 \pi r(h+r) m^{2}

=2 \times \frac{22}{7} \times 7(15+7) \mathrm{m}^{2}

=2 \times \frac{22}{7} \times 7 \times 22 \mathrm{~m}^{2}

=968 \mathrm{~m}^{2}

Therefore, a total of 968 \mathrm{~m}^{2} of the area needs painting.

Calculating the cost of painting 968 \mathrm{~m}^{2} area of cylindrical tin at ₹ 2 / m^{2}

= ₹ 968 x 2

= ₹1936

Hence, Ishan’s parents will have to spend a total of ₹1936 to paint one cylindrical tin to decorate his study area.

Question 2: Find the curved surface area of a cylinder whose diameter is 14 cm and height is 15 cm.

Solution:

Radius of cylinder (\mathrm{r})=\frac{14}{2} \mathrm{~cm}=7 \mathrm{~cm}

Height of cylinder (h) = 15 cm

CSA = 2πrh

=2 \times \frac{22}{7} \times 7 \times 15 \mathrm{~cm}^{2}

=660 \mathrm{~cm}^{2}

Therefore, the curved surface area of a cylinder =660 \mathrm{~cm}^{2}.