A cylinder is a three-dimensional structure having a circular base. A cylinder can be seen as a set of circular disks that are stacked on one another. 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. Total surface area of a cylinder is equal to the sum of areas of all its faces. Let us now find out a general formula for the calculation of the surface area of a cylinder.

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 height ‘h’ units. Thus,

Curved surface area of a cylinder of base radius ‘r’, and height ‘h’ = 2𝝿 × r × h

Total surface area of a cylinder of base radius ‘r’, and height ‘h’ = 2𝝿 × r × h + area of two circular bases = 2𝝿 × r × h + 2𝝿r^{2}= 2𝝿r (h + r)

### Problems related to the calculation of the surface area of a cylinder:

Question: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 𝝿 = 22/7)

Solution:Total surface area of aquarium = 2𝝿r (h + r)=\(= 2 \times \frac{22}{7} \times 7 \times 20\)^{2}

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

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