Surface Area of a Right Circular Cylinder
If we take a number of circular sheets of paper and stack them up as we stacked up rectangular sheets earlier,
what would we get (see Fig. 13.6)?
Here, if the stack is kept vertically up, we get what is called a right circular cylinder, since it has been kept at right angles to the base, and the base is circular.
Let us see what kind of cylinder is not a right circular cylinder.
In Fig 13.7 (a), you see a cylinder, which is certainly circular, but it is not at right angles to the base.
So, we can not say this a right circular cylinder. Of course, if we have a cylinder with a non circular base, as you see in Fig. 13.7 (b), then we also cannot call it a right circular cylinder.
`color{red}ul("Remark :")` Here, we will be dealing with only right circular cylinders. So, unless stated otherwise, the word cylinder would mean a right circular cylinder.
Now, if a cylinder is to be covered with coloured paper, how will we do it with the minimum amount of paper?
First take a rectangular sheet of paper, whose length is just enough to go round the cylinder and whose breadth is equal to the height of the cylinder as shown in Fig. 13.8.
The area of the sheet gives us the curved surface area of the cylinder. Note that the length of the sheet is equal to the circumference of the circular base which is equal to `2πr.`
So, curved surface area of the cylinder
`color{green}(= "area of the rectangular sheet = length × breadth")`
`color{orange}(= "perimeter of the base of the cylinder × h")`
`color{red}(= 2πr × h)`
Therefore
where r is the radius of the base of the cylinder and h is the height of the cylinder.
`color{red}("Remark :")` In the case of a cylinder, unless stated otherwise, ‘radius of a cylinder’ shall mean’ base radius of the cylinder’.
If the top and the bottom of the cylinder are also to be covered, then we need two circles (infact, circular regions) to do that, each of radius r, and thus having an area of `πr^2` each (see Fig. 13.9), giving us the total surface area as `color{navy}(2πr^2 + 2πrh = 2πr(r + h).)`
So,
where h is the height of the cylinder and r its radius.
`color{red}("Remark :")` You may recall from Chapter 1 that π is an irrational number. So, the value of π is a non-terminating, non-repeating decimal. But when we use its value in our calculations, we usually take its value as approximately equal to `(22)/7` or `3.14.`
what would we get (see Fig. 13.6)?
Here, if the stack is kept vertically up, we get what is called a right circular cylinder, since it has been kept at right angles to the base, and the base is circular.
Let us see what kind of cylinder is not a right circular cylinder.
In Fig 13.7 (a), you see a cylinder, which is certainly circular, but it is not at right angles to the base.
So, we can not say this a right circular cylinder. Of course, if we have a cylinder with a non circular base, as you see in Fig. 13.7 (b), then we also cannot call it a right circular cylinder.
`color{red}ul("Remark :")` Here, we will be dealing with only right circular cylinders. So, unless stated otherwise, the word cylinder would mean a right circular cylinder.
Now, if a cylinder is to be covered with coloured paper, how will we do it with the minimum amount of paper?
First take a rectangular sheet of paper, whose length is just enough to go round the cylinder and whose breadth is equal to the height of the cylinder as shown in Fig. 13.8.
The area of the sheet gives us the curved surface area of the cylinder. Note that the length of the sheet is equal to the circumference of the circular base which is equal to `2πr.`
So, curved surface area of the cylinder
`color{green}(= "area of the rectangular sheet = length × breadth")`
`color{orange}(= "perimeter of the base of the cylinder × h")`
`color{red}(= 2πr × h)`
Therefore
`color{red}("Curved Surface Area of a Cylinder" = 2πrh)`
where r is the radius of the base of the cylinder and h is the height of the cylinder.
`color{red}("Remark :")` In the case of a cylinder, unless stated otherwise, ‘radius of a cylinder’ shall mean’ base radius of the cylinder’.
If the top and the bottom of the cylinder are also to be covered, then we need two circles (infact, circular regions) to do that, each of radius r, and thus having an area of `πr^2` each (see Fig. 13.9), giving us the total surface area as `color{navy}(2πr^2 + 2πrh = 2πr(r + h).)`
So,
`color{green}("Total Surface Area of a Cylinder" = 2πr(r + h))`
where h is the height of the cylinder and r its radius.
`color{red}("Remark :")` You may recall from Chapter 1 that π is an irrational number. So, the value of π is a non-terminating, non-repeating decimal. But when we use its value in our calculations, we usually take its value as approximately equal to `(22)/7` or `3.14.`
