3-D Cylinder & its Formulas :
A cylinder is one of the common shapes we see in everyday life. It is three dimensional(3-D). It has two parallel faces which are called as bases or ends of the cylinder.
A curved surface joins these two bases. Therefore, a cylinder area, more predominantly referred as surface area of cylinder, includes two types of areas.
One is the area of the bases, called as base areas, and the other is the area of the joining curved surface. The latter is better known as the lateral surface of a cylinder. In general surface area of cylinder refers to the sum of lateral area and the areas of the two bases.
Let us illustrate how the formula for surface area of a cylinder can be derived in a simple way.
In the above diagram, a cylinder of radius r and height h is shown on the left.
Suppose the cylinder (considering it to be hollow) is ripped along the dotted line shown and opened on both the directions, we can see a rectangle formed as shown on the right. The length of the rectangle is equal to the circumference of the end circles
which is 2πr,
because the entire circumference is opened out .The height of the rectangle remains same as the height of the cylinder as h.
It can be seen as obvious that the area of the curved surface of the cylinder is same as the area of the rectangle, which is
2πr* h = 2πrh.
Therefore, the lateral surface area of a cylinder is given by
L = 2πrh.
The total surface area is the sum of the lateral surface area and the areas of the two bases which are circular.
The areas of the two bases are the areas two circles which is
πr^2 + πr^2 = 2πr^2.
Therefore, the total surface area of the cylinder is,
(2πr^2 + 2πrh).
The same can be simplified in factored form as
2πr(r + h).
Now let us look into the significance of surface area and lateral area of a cylinder. The surface area tells us the amount of surface, in turn, the area of the sheet metal to that is needed to fabricate a cylinder.
In many cases, cylindrical shapes in large sizes without ends are used as shells (think of water or any fluid pipe lines). Therefore, in such cases, only the lateral surface areas of cylinders play the role.
There are some more interesting facts about the cylinder areas and cylinder volumes. Suppose you are asked to design a hollow cylinder for a given volume that involves minimum cost to fabricate. That is, to find the dimensions with least surface area.
Mathematically we can prove that for a fixed volume of a cylinder, the least surface area can be achieved when h = 2r, that is when the height of the cylinder equals the diameter of the cylinder.
Same way, for a given surface area of a cylinder, maximum volume can be obtained by making the height and diameter of the cylinder congruent.