Mensuration is the study of various dimensions of the different geometrical shapes in mathematics. People come across geometrical shapes not only in mathematical theory but in our daily lives as well. Measuring their dimensions, shaping new objects in a particular fashion, shape, size, etc. comes under the scope of mensuration. Designing a simple lunch box also falls under the scope of mensuration. Constructing buildings, dams, etc. everything would require the study of geometrical shapes.
Basic Terminology
In mensuration, there are some basic terminologies that are used to define the different characters of different shapes. They are area, perimeter, volume, curved surface area, total surface area, etc. Let’s take a look at their definitions in detail,
- Area: In simple words, the area is the parameter which helps to quantify the region occupied by a shape.
- Curved Surface Area: The curved surface area of a shape would take into account only the curved surface and not the base or top of such shape.
- Total Surface Area: The total surface area of a shape is described as the area bounded by the whole shape. It takes into account the curved surface area as well as the areas of the base and the top.
- Volume: Volume is the quantification of the space which is contained inside a shape. Thus the volume of a shape indicates how much storage it can take up.
- Perimeter: Perimeter is the parameter that helps quantify the region around a given shape.
Cylinder
Such a three-dimensional shape that is formed by joining two circular bases, conjoined by a curved surface in between is called a cylinder. A cylinder, when seen from the top looks like a circle or ellipse, depending upon the shape of the given bases, and rectangular when viewed from the side. It is to be noted that a cylinder is hollow from the inside, and as such can be used to store stuff. The amount of stuff which can be stored in it is depicted by its capacity/ volume. Another important characteristic to be noted is that the bases of a cylinder must always be congruent and parallel. Such a cylinder only has one base and whose curved surface heights intersect at its top forms a right-angled cone. Hence a cone is a special case of a cylinder.
The radii of the circular top and the base is termed as the radius of the cylinder itself and the distance between the circular top and bottom is termed as the height of the cylinder. The following figure shows a cylinder with its circular bases and the curved surface joining them,
The top and side view of a cylinder is depicted below:
Properties of a Cylinder
- A cylinder has three dimensions, namely, its two circular bases and height.
- The top view of a cylinder is a circle.
- The side view of a cylinder is that of a rectangle.
- The different measures associated with a cylinder are its lateral surface area, volume, and its total surface area.
- The bases of a cylinder are always parallel.
- A cylinder has the same cross-sections as that of a prism.
- An elliptical cylinder is the one whose bases are not circular, but elliptical in shape.
- A right circular cylinder is formed when the bases are circular in shape and form a right angle with the axis.
Curved Surface Area of a Cylinder
The curved surface area of a cylinder is defined as the area which is contained between the two parallel circular bases. This means that for the calculation of the curved surface area, the areas of the circular bases are not included.
The curved surface area of a cylinder = 2πrh unit2, where r is the radius of the cylinder and h is the height, i.e., the distance between the parallel bases.
Find the curved surface area of a cylinder whose radius is 4 cm and height 8 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 4 cm and h = 8 cm. Substituting the given values in the formula,
CSA = 2π(4)(8) cm2
= 64 (3.14) cm2
= 200.96 cm2 (assuming, π = 3.14)
Hence the curved surface area of the cylinder is 200.96 sq. units.
Similar Problems
Question 1: Find the curved surface area of a cylinder given its height is 10 cm and radius 5 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 5 cm and h = 10 cm. Substituting the given values in the formula,
CSA = 2π(5)(10) cm2
= 100 (3.14) cm2
= 314 cm2 (assuming π = 3.14)
Hence the curved surface area of the cylinder is 314 cm2.
Question 2: Find the curved surface area of a cylinder given its radius of 6 cm and height of 9 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 6 cm and h = 9 cm. Substituting the given values in the formula,
CSA = 2π(6)(9) cm2
= 108 (3.14) cm2
= 339.12 cm2 (assuming π = 3.14)
Hence the curved surface area of the cylinder is 339.12 cm2.
Question 3: Find the curved surface area of a cylinder given its height of 100 cm and radius of 10 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 10 cm and h = 100 cm. Substituting the given values in the formula,
CSA = 2π(10)(100) cm2
= 1000 (3.14) cm2
= 3140 cm2 (assuming π = 3.14)
Hence the curved surface area of the cylinder is 3140 cm2.
Question 4: Find the curved surface area of a cylinder given its height 14 cm and radius 30 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 30 cm and h = 14 cm. Substituting the given values in the formula, we get
CSA = 2π(30)(14) cm2
= 840 (3.14) cm2
= 2637.60 cm2 (assuming π = 3.14)
Hence the curved surface area of the cylinder is 2637.60 cm2.
Question 5: Find the curved surface area of a cylinder given its height of 8 cm and radius of 35 cm.
Solution:
Curved surface area of a cylinder = 2πrh unit2
Here, r = 35 cm and h = 8 cm. Substituting the given values in the formula,
CSA = 2π(35)(8) cm2
= 560 (3.14) cm2
= 1758.40 cm2 (assuming π = 3.14)
Hence the curved surface area of the cylinder is 1758.40 cm2.