How do I calculate the volume of a cylinder if only the surface and curved area is given?
A cylinder is one of the most common regular geometric shaped objects that has wide scale application. It has conventionally been a three-dimensional solid, which is one of the most fundamental curvilinear geometric shapes. In terms of geometry, it can be thought to be a prism with a circular base. This customary view is utilized in the elementary forms of geometry, but from an advanced mathematical viewpoint, it has moved to the infinite curvilinear surface. This transition from the conventional to modern has fashioned some ambiguity with its terminology. Both these views are imagined and distinguished from each other with reference to solid cylinder and cylindrical surfaces. For all sense and purpose, our discussion will be focused on a right circular cylinder. The following figure below, is a simple representation of a right circular cylinder with the usual representations.
Here, ‘h’ is the height of this cylinder from the base to the top and ‘r’ is the radius of the flat circular bases. Now, let us understand how a right circular cylinder can be created. By having a look at the figure, the right cylinder can be assumed to be a series of similar shaped circles or radius r, stacked one upon the other till a height of ‘h’ is reached. This will create a solid cylinder. In order to create a hollow right circular cylinder, it can be assumed to be a rectangle having a length ‘l’ and width of ‘h’. This rectangle is folded along its length to form a circle with a radius ‘r’, such that Thus, this radius can be calculated as . This cylinder is capped on the bottom and top by a circle base of the same radius ‘r’.
Various parameters can be defined for this cylinder, such as the curved surface area, total surface area, volume of the cylinder etc. Let us discuss each of these properties one by one. Curved surface area is defined as the total surface area that is occupied by the curved portion of the cylinder alone. This does not include the surface area of the top and bottom circular bases. With the usual notation, the curved surface area (CSA) is calculated as follows
The total surface area of the cylinder is defined as the summation of the curved surface area and the surface area of the top and bottom circular bases. Using the same notations, total surface area can be calculated as follows,
This can be further simplified as
The volume of the cylinder is defined as the total space enclosed by the curvilinear geometry of this shape. This is calculated as
Now, the problem at hand is how to calculate the volume of a cylinder if only the total curved area and the total surface area of the cylinder are given to us. Starting from the formulae available for total surface area and curved surface area, let us derive an expression for the same. Let us assume curved surface area is CSA and the total surface area is given by TSA. Taking the ratio of the curved surface area to total surface area, we obtain the following,
From the ratio, it is clear that the term is common in the numerator and denominator. Thus, this term can be cancelled out. After cancelling the common term, we obtain the following,
On simplifying the above ratio, we will try to express the height of the cylinder ‘h’ in terms of its radius. Thus,
- TSA = (h +r ) CSA
thus, (TSA-CSA) h = CSA. r
From here, it can be concluded that if only the total surface area and total curved area of a cylinder are given, then the relation between the height and the radius can be drawn to be as follows,
Height of the cylinder ‘h’ = [CSA/(TSA-CSA)].r
It should be clear that the total surface area will always be larger than the total curved surface area
Inputting the value of h in the curved area expression, we can get,
Thus, the radius of the cylinder can be determined as and the height is calculated as
Therefore, the volume of a cylinder for which only TSA and CSA are given is calculated as
Inserting the expressions of r and h,
Let us assume a right circular cylinder having the usual notation ‘r’ for its radius and ‘h’ for its height. The value of the total surface area is 200 sq. cm and the total curved area of this cylinder is 160 sq. cm. Determine the volume, the height and the radius of this cylinder.
Substituting the values we get, volume is 201.85 cubic cm, height is 10.09 cm and radius is 2.52 cm.
More example questions: Indefinite Integrals from Class 12 Maths
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