Right Circular Cylinder is a three-dimensional solid figure. The right circular cylinder has a closed circular surface on both ends, which are parallel to each other. A right circular cylinder is also known as the right cylinder. From the axis of the right circular cylinder, all the points that lie on the closed surface are equidistant. A right circular cylinder is the most commonly used \(3D\) figure in our real-life.

In this article, we will learn everything about right circular cylinders including surface area, volume and properties.

What is a Right Circular Cylinder?

A cylinder is a solid geometrical figure with straight parallel sides and a circular or oval cross-section. You can understand a cylinder as multiple circular disks stacked on top of each other.

The radius of the right circular cylinder so formed is equal to the radius of the circular disk. The number of disks determines the height of the cylinder.

The cylinder has three faces, one curved face and two flat faces, two edges (where two faces meet) and zero vertices (corners where two edges meet).

How is a Right Circular Cylinder Formed?

The right circular cylinder is a three dimensional solid. A right circular cylinder is an object formed by rolling the rectangle on one of its sides, taking as an axis of a cylinder.

If the axis (one of the sides of the rectangle) is perpendicular to the radius \((r)\), then the cylinder, so formed, is called a right circular cylinder.

The right circular cylinder has a base, and the top are circular in shape and are parallel to each other. The distance between these circular ends of a cylinder is known as the height \((h)\) of a cylinder.

Properties of a Right Circular Cylinder

The right circular cylinder is a three-dimensional figure, which is formed by using two circles and one rectangle. Every solid in the geometry has its own properties. Similarly, the properties associated with the cylinder are given below:

1. A cylinder has two curved edges, one curved surface and two flat circular faces.

2. The two circular flat surfaces of the cylinder are always congruent to each other. The top circular base and bottom circular base of the cylinder is the same.

3. The centre line of the cylinder is known as the axis of the cylinder. The axis of the cylinder forms a right angle with the bases.
4. A cylinder does not have any vertex or specific corner.
5. From the axis of the right circular cylinder, all the points that lie on the closed surface are equidistant.
6. The cylinder size depends on the measure of the radius of the base and height of the cylinder.

Surface Area of a Right Circular Cylinder

The area covered by the surfaces of the cylinder is known as the surface area of the cylinder. There are two types of areas, which are given below:

1. Curved surface area (C.S.A)
2. Total surface area (T.S.A)

Curved Surface Area of a Cylinder

It is the area covered by the curved face of the cylinder. It is also known as the lateral surface area of the cylinder. The curved surface area of the cylinder does not include the area of the bases.

As we know, a cylinder is formed by rolling a rectangle to one of its sides. So, the curved surface of the cylinder is the area covered by a rectangle of length \(2πr\) and width \(h\).

So, the area of the rectangle is \({\text{Length}} \times {\text{width}} = 2\pi r \times h\)

\({\text{Curved}}\,{\text{surface}}\,{\text{area}}\,\left({{\text{C}}{\text{.S}}{\text{.A}}} \right) = 2\pi rh\)

Total Surface Area of a Cylinder

The total surface area of the cylinder is equal to the sum of the area of the curved surface and the area of the circular bases.

Let us consider a cylinder with radius \(r\), and height \(h\). On rolling the cylinder as shown in the below image:

After complete unroll, the cylinder looks like three parts, such as a rectangle and two circular surfaces.

From the above, we can say that the total surface area of the cylinder is a sum of the area of the rectangle and the area of two circular bases.

Total surface area (T.S.A) of the cylinder \( = 2\pi rh + \pi {r^2} + \pi {r^2} = 2\pi rh + 2\pi {r^2}\)

\({\text{Total}}\,{\text{surface}}\,{\text{area}}\,\left( {{\text{T}}{\text{.S}}{\text{.A}}} \right) = 2\pi r\left( {r + h} \right)\)

The units of both curved surface area and total surface area of the cylinder are measured in square units like \({\text{c}}{{\text{m}}^{\text{2}}}{\text{,}}\,{{\text{m}}^{\text{2}}}{\text{,}}\,{\text{sq}}{\text{.ft}}\) etc.

Volume of a Right Circular Cylinder

The volume of the object in a three-dimensional geometry is the space occupied by the matter (gas, liquid, and solid). In general, the capacity of the object is the volume of the object. Thus, the volume of the cylinder is the amount of space occupied by a matter inside a cylinder. 

The volume of the cylinder is equal to the product of the area of the circular base and the height of the cylinder.

\({\text{Volume}}\,{\text{=}}\,{\text{Area}}\,{\text{of}}\,{\text{the}}\,{\text{base}}\,{\text{ \times }}\,{\text{height}}{\text{.}}\)

Let the cylinder of radius \(‘r’\) and height \(‘h’\)  as shown below:

We know that area of a circle with radius \(r\) is \({r^2}\).

Therefore, the volume of the cylinder is \(\pi {r^2} \times h = \pi {r^2}h\).

\({\text{Volume}} = \pi {r^2}h\)

In general, the units of the volume of the cylinder is measured in cubic units like cubic meters \(\left({{{\text{m}}^3}}  \right)\), cubic centimetres \(\left({{{\text{cm}}^3}}  \right)\), cubic millimetres \(\left({{{\text{mm}}^3}}  \right)\), etc.

Examples of Right Circular Cylinder

The right circular cylinder is the most used solid objects in real life. Gas cylinder, candles, water bottles are some of them are listed below:

Important Formulas for Right Circular Cylinder

Formulas for Hollow Cylinder

Projective Geometry

In projective geometry, a cylinder is simply a cone whose apex (vertex) lies on the plane at infinity. If the cone is a quadratic cone, then the plane at infinity (which passes through the vertex) can intersect the cone with two real lines, a single real line (actually a conjugate pair of lines), or only at the vertex. These cases give rise to hyperbolic, parabolic or elliptical cylinders, respectively.

This concept is useful when considering degenerate cones, which may include cylindrical cones.

Prism

A solid circular cylinder can be viewed as the finite case of an n-gonal prism where n approaches infinity. The connection is very strong and many older texts treat prism and cylinder together. The formulas for surface area and volume are obtained from the respective formulas for prisms by using inscribed and enclosed prisms and then allowing the number of sides of the prism to increase without bounding.

One reason for the initial emphasis (and sometimes exclusive treatment) of circular cylinders is that a spherical base is the only type of geometric figure for which this technique works with the use of only elementary ideas (calculus or any for the more advanced maths) The terminology about prisms and cylinders is the same. Thus, for example, since a truncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes is called a truncated cylinder.

From a polyhedron point of view, a cylinder can also be viewed as a dual bicone of an infinitely sided form of the bipyramid.

Cylindrical Surface

In some areas of geometry and topology, the term cylinder refers to what is called a cylindrical surface. A cylinder is defined as a surface consisting of all points on all lines that are parallel to a given line and which pass through a certain plane curve in a plane that is not parallel to the given line. Huh. Such cylinders are sometimes referred to as generalised cylinders. A unique line passes through each point of a normalized cylinder that lies in the cylinder. Thus, this definition can be redefined to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines.

A cylinder that has a right segment that is an ellipse, parabola, or hyperbola is called an elliptical cylinder, parabolic cylinder, and hyperbolic cylinder, respectively. These are degenerate quadrilateral surfaces. When the principal axes of a quadric are aligned with a reference frame (always possible for quadrics), a general equation of a quadric in three dimensions is given by a parabolic cylinder equation:

Solved Examples – Right Circular Cylinder

Q.1. The radius of a cylindrical milk bottle is \(2\,{\text{cm}}\), and whose height is \(5\,{\text{cm}}\). Find the capacity of milk in the bottle? (Use: \(\pi = \frac{{22}}{7}\))

Ans: Given the radius of the bottle \(\left( r \right) = 2\,{\text{cm}}\) and height \(\left( h \right) = 5\,{\text{cm}}\).
We know that the capacity of the bottle is known as the volume of the cylinder.The volume of the cylinder with radius \(‘r’\) and height \(‘h’\) is \(\pi {r^2}h\).
\( = \frac{{22}}{7} \times {2^2} \times 5\)
\( = \frac{{22}}{7} \times 20\)
\( = \frac{{440}}{7}\)
\( = 62.86\,{\text{c}}{{\text{m}}^3}\)
Hence, the capacity of the milk bottle is \( = 62.86\,{\text{c}}{{\text{m}}^3}\) (approx)

Q.2. The volume of a cylindrical water tank is \(1100\,{{\text{m}}^3}\) And the radius of the base of the cylindrical tank is \(5\,{\text{m}}\). Calculate the height of the tank.
Ans: Given, the radius of the cylinder \(\left( r \right) = 5\,{\text{m}}\) and volume of the tank is \(1100\,{{\text{m}}^3}\).
We know that volume of the cylinder with radius \(‘r’\) and height \(‘h’\) is \(\pi {r^2}h\).
\( \Rightarrow 1100 = \frac{{22}}{7} \times {(5)^2} \times h\)
 \( \Rightarrow 1100 = \frac{{22 \times 25}}{7}h\)
 \( \Rightarrow 1100 = \frac{{550}}{7}h\)
 \( \Rightarrow h = 1100 \times \frac{7}{{550}}\)
 \( \Rightarrow h = 2 \times 7\)
 \( \Rightarrow h = 14\,{\text{m}}\)
Hence, the height of the water tank is \(14\,{\text{m}}\).

Q.3. Find the total surface area of the container, which is in the shape of a cylinder, whose radius and height are \(7\,{\text{cm}}\) and \(10\,{\text{cm}}\), respectively.
Ans: Given the radius of the container \(\left(r \right) = 7\,{\text{cm}}\) and height \(\left(h \right) = 10\,{\text{cm}}\).

The total surface area of the cylinder with radius \((r)\) and height \((h)\) is \(2πr(r+h)\)
\(=2 \times \frac{22}{7} \times 7 \times(7+10)\)
\( = 2 \times 22 \times 17\)
\( = 748\,{\text{c}}{{\text{m}}^2}\)
Therefore, the total surface area of the container is \(748\,{\text{c}}{{\text{m}}^2}\).

Q.4. Calculate the cost required to paint a cylindrical tank, whose base radius is \(7\,{\text{m}}\), and height \(13\,{\text{m}}\). If the cost of painting the tank is \(₹ 2\) per \({{\text{m}}^2}\).
Ans: Given, the radius of the tank \(\left(r \right) = 7\,{\text{m}}\) and height \(\left(h \right) = 13\,{\text{m}}\).
The total surface area of the cylinder with radius \((r)\) and height \((h)\) is \(2πr(r+h)\).
\(=2 \times \frac{22}{7} \times 7 \times(7+13)\)
\(=2 \times 22 \times 20\)
\( = 880~{{\text{m}}^2}\)
Given, the cost of painting the tank is \(₹ 2\) per \({{\text{m}}^2}\).
So, the total cost of painting the tank is given by \(₹ 2 \times 880=₹ 1760\).
Hence, the total cost of painting the tank is \(₹ 1760\).

Q.5. The volume of the cylinder is \(220\,{{\text{m}}^3}\). And, the height of the cylinder is \(10\,{\text{m}}\), then find the base radius of the cylinder?
Ans: Let the base radius of the cylinder is \(r m\).
Given: Volume of the cylinder is \(220\,{{\text{m}}^3}\) and height of the cylinder \(\left( h \right) = 10\,{\text{m}}\).
We know that volume of the cylinder with radius \(‘r’\) and height \(‘h’\) is \(\pi {r^2}h\).
\(\Longrightarrow 220=\frac{22}{7} \times(r)^{2} \times 10\)
\(\Longrightarrow 220=\frac{220}{7} \times(r)^{2}\)
\(\Rightarrow r^{2}=7\)
\(\Rightarrow r=\sqrt{7}\)
Hence, the base radius of the cylinder is \(\sqrt 7 \,{\text{m}}\).

Summary

In this article, we have studied the cylinder and right circular cylinder and its properties. We also learnt the formulas of surface area and volume of the right circular cylinder, which help us to solve real-life problems. The real-life examples of the right circular cylinder given in this article help you in recognising the cylindrical objects around us easily.

FAQs

Q.1. Why a cylinder is called a right circular cylinder?
Ans: The right circular cylinder has a closed circular surface on both ends, which are parallel to each other. 
Since the line joining centres of two bases are perpendicular to the plane of bases, so it is called the right circular cylinder.

Q.2. What is the formula of CSA of the right circular cylinder?
Ans: We know that the C.S.A of a cylinder is known as the curved surface area of the cylinder. For the cylinder with radius \(‘r’\) and height \(‘h’\), the C.S.A is \(2πrh\).

Q.3. What is the difference between a cylinder and a right circular cylinder?
Ans: The cylinder is three-dimensional solid, having one curved surface and two flat circular surfaces.
The right circular cylinder is the cylinder, in which the axis of the cylinder is making right angles to the side.

Q.4. Why is the volume of a cylinder formula is \(\pi {r^2}h\)?
Ans: The volume of a cylinder is equal to the product of the area of the circular base and the height of the cylinder.
Volume = Area of the base \( \times \) height
We know that area of a circle with radius \(r\) is \({r^2}\).
Therefore, the volume of the cylinder is \(\pi {r^2} \times h = \pi {r^2}h\).

Q.5. What is the formula of TSA of the right circular cylinder?
Ans: We know that T.S.A of the cylinder is known as the total surface area of the cylinder, which is the sum of the area of curved surface and area of the bases. For the cylinder with radius \(‘r’\) and height \(‘h’\), the total surface area is \(2πr(r+h)\).

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