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. Since the cylinder is a three-dimensional shape, 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. Let us learn here how to derive the surface area for any given cylinder along with the formula and solved examples.
Surface 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 (CSA) is defined as the area of the curved surface of any given cylinder having 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;
|CSA or LSA = 2π × r × 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 ‘r’, and height ‘h’ is equal to the curved area and circular areas of the cylinder.
|TSA = 2π × r × h + 2πr2= 2πr (h + r) Square units|
Surface Area of Cylinder Derivation
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 ‘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,
By the formula of area of the circle, we know,
Area of circular base of cylinder = πr2
Since, there are two circular bases, therefore area of both the circular bases = πr2+πr2 = 2πr2 ……………….(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 x πd = h x 2πr ( 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 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.
Surface Area of a Cylinder Problem
Q.1: 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/m2. (Take π = 22/7)
Total surface area of aquarium = 2πr (h + r)= 2 x 22/7 x 7 x 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 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 = 2πr (h + r) = 2x 22/7 x 14 x (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.
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