What is the total surface area and curved surface area of cylinder?

The surface area of a cylinder is the area occupied by its surface in a three-dimensional space. A cylinder is a three-dimensional structure having circular bases which are parallel to each other. It does not have any vertices. Generally, the area of the three-dimensional shapes refers to the surface area. The surface area is represented in square units. For example, cm2, m2, and so on. A cylinder can be seen as a set of circular disks that are stacked on one another.  Since the cylinder is a solid of a three-dimensional shape,  it has both surface area and volume. 

The cylinder area is defined as the sum of the curved surface and the area of two circular bases of the cylinder.

As we know, a cylinder has two types of surfaces, one is the curved surface and the other is the circular base. So the total surface area will be the sum of the curved surface and two circular bases.

Also, read: volume of cylinder.

Table of contents:

Cylinder Area Definition

The area of the cylinder is the total region covered by a cylinder in three-dimensional space. The cylinder area is equal to the sum of the area of two circular bases and the curved surface area. In right cylinders, the two circular bases are exactly over each other and the axis line produces a right angle to the base. In case one of the circular bases is displaced and the axis does not produce the right angle to the base, then it is called the oblique cylinder.

In the middle of the two circular bases, there is a curved surface, which, when opened represents a rectangular shape. This curved surface is also called a lateral surface. The different parameters that are used to calculate the cylinder area include radius, height, axis, base, and side. The radius of the cylinder is defined as the radius of the circular base. The height of the cylinder is calculated by measuring the perpendicular distance between two circular bases, and the line that joins the centre of the base is called the axis. 

Area of a Cylinder Formula

The total area of a cylinder is based on:

• Curved Surface Area (CSA)
• Base Area

Curved Surface Area

The curved surface area of a cylinder (CSA) is defined as the area of the curved surface of any given cylinder having a base radius ‘r’, and height ‘h’, It is also termed as Lateral Surface Area (LSA). The formula for a curved area or lateral area is given by:

Base Area of Cylinder

The base of the cylinder is a circular shape. Hence, by the formula of area of the circle, we know,

Area of the circular bases of cylinder = 2 (πr2)  [Since the cylinder has two circular bases]

Total Surface Area of Cylinder

The total surface area of a cylinder is equal to the sum of the areas of all its faces. The total surface area with radius ‘r’, and height ‘h’ is equal to the sum of the curved area and circular areas of the cylinder.

Also, read:

Derivation of Surface Area of Cylinder

Now, think of a scenario where we need to paint the faces of a cylindrical container. Before we start painting, we need to know the quantity of paint required for painting all the walls. Thus, we need to find the area of all the faces of this container to calculate the amount of paint needed. We define this term as the total surface area.

Let us take a cylinder of base radius ‘r’ and height ‘h’ units. The curved surface of this cylinder, if opened along the diameter (d = 2r) of the circular base can be transformed into a rectangle of length ‘2πr’ and breadth ‘h’ units.

By the formula of area of the circle, we know,

Area of the circular base of cylinder = πr2

Since there are two circular bases, therefore the area of both the circular bases = πr2 + πr2 = 2πr2 ……………….(1)

Now, from the figure, you can see, that when we open the curved surface of the cylinder in two-dimension space, it forms a rectangle. Hence, the height and circumference of the circular bases are the dimensions of the rectangle formed from it. Therefore,

Area of the curved surface = Height × Circumference

Curved Surface area = h ×  πd = h × 2πr (since d = 2r)

CSA = 2πrh …………….(2)

By adding equation 1 and equation 2, we get the total surface area, such that;

Total Surface area = Curved Surface area + Area of Circular bases

TSA = 2πrh + 2πr2

By taking 2πr as a common factor from RHS, we get;

TSA = 2πr (h + r)

This is the formula for the total surface area of a given cylinder whose radius is r and height is h.

Problems and Solutions

Q.1: Calculate the cost required to paint a container which is in the shape of a right circular cylinder having a base radius of 7 m and a height of 13 m. If the painting cost of the container is INR 2.5/m2. (Take π = 22/7)

Solution:

Total surface area of aquarium = 2πr (h + r)= 2 × (22/7) × 7 × 20 = 880 m2

Total cost of painting the container = 2.5 × 880 = Rs. 2200

Q.2: Find the total surface area of a container in a cylindrical shape whose diameter is 28 cm and height is 15 cm.

Solution:

Given, diameter = 28 cm, so radius = 28/2 = 14 cm

and height = 15 cm

By the formula of total surface are, we know;

TSA = 2πr (h + r) = 2 × (22/7) × 14 × (15 + 14)

TSA = 2 × 22 × 2 × 29

TSA = 2552 sq.cm

Hence, the total surface area of the container is 2552 sq. cm.

Practice Questions

1. A water tank has a radius of 40 inches and a height of 150 inches. Find the area. 
2. Find the radius of a cylinder whose total surface area is 2136.56 square cm and the height is 3 cm.
3. Find the curved surface area of a cylinder whose diameter is 56 cm and height is 20 cm.

To learn and practise more problems related to the calculation of surface area and volume of a cylinder, download BYJU’S – The Learning App.