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. A cylinder can be seen as a set of circular disks that are stacked on one another.Â Since the cylinder is a three-dimensional shape,Â it has both surface area and volume.

As we know, a cylinder has two types of surfaces, one is the curved surface and the other is the circular bases. So the total surface area will be the sum of the two. Even to find the volume of cylinder, we consider both circular and lateral surfaces. Let us learn here how to derive the surface area for any given cylinder along with the formula and solved examples.

**Table of contents:**

## Cylinder Area

The area of the cylinder is the total region covered by a cylinder in three-dimensional space. It is basically equal to the sum of area of two circular bases and curved surface area. The area for both the bases are equal for both right cylinder and 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 lateral surface. Now let us find the total cylinder area using formulas.

## Area of a Cylinder Formula

The **surface area of a cylinder** can be classified into two types namely,

- Curved Surface Area (CSA)
- Total Surface Area (TSA)

### Curved Surface Area

The **curved surface area of a cylinder**Â (CSA) is defined as the area of the curved surface of any given cylinder having baseÂ radius â€˜râ€™, and height â€˜hâ€™, It is also termed asÂ **Lateral surface area (LSA)**. The formula for a curved area or lateral area is given by;

CSA or LSA = 2Ï€ Ã— r Ã— h Square units |

### Total Surface Area of Cylinder

The** total surface area of a cylinder** is equal to the sum of areas of all its faces. The Total surface area with radius â€˜râ€™, and height â€˜hâ€™ is equal to the curved area and circular areas of the cylinder.

TSA = 2Ï€ Ã— r Ã— h + 2Ï€r^{2}= 2Ï€r (h + r) Square units |

**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 know the area of all the faces of this container in order to calculate the quantity of paint required. We define this term as the total surface area.

Let us take a cylinder of base radius â€˜râ€™ and height â€˜hâ€™ 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 â€˜2Ï€râ€™ and height â€˜hâ€™ units. Thus,

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

Area of circular base of cylinder =Â Ï€r^{2}

Since, there are two circular bases, therefore area of both the circular bases =Â Ï€r^{2}+Ï€r^{2} = 2Ï€r^{2}Â ……………….(1)

Now, from the figure you can see, 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 x Circumference

Curved Surface area = h xÂ Â Ï€d = h x 2Ï€r ( d = 2r)

CSA =Â 2Ï€rh …………….(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 =Â 2Ï€rh +Â 2Ï€r^{2}

By taking 2Ï€r as common factor from RHS, we get;

**TSA = 2Ï€r (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:Â Calculate the cost required to paint a container which is in shape of a right circular cylinder having a base radius of 7 m and height 13 m. If the painting cost of the container is INR 2.5/m ^{2}. (Take Ï€ = 22/7)**

**Solution:**

Total surface area of aquarium = 2Ï€r (h + r)= 2 x 22/7 x 7 x 20Â = 880 m^{2}

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

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

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

and height = 15cm

By the formula of total surface are, we know;

TSA =Â 2Ï€r (h + r) =Â 2x 22/7 xÂ 14 xÂ (15 + 14)

TSA = 2 x 22 x 2 x 29

TSA = 2552 sq.cm

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

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

thanks for your help it really helped me get an A*