# Cylinder: Properties, Cylinder Volume and Area Formulas

We know of several things that are cylindrical in shape from our day-to-day life. A cylinder or a cylindrical structure is traditionally considered as a three-dimensional solid in the shape of a prism with a circle at its base. It is one of the most basic curvilinear geometric shapes. This traditional view is still useful in solving elementary geometric problems. But the advanced mathematical viewpoint is that a cylindrical surface is an infinite curvilinear surface. This definition is currently used in various modern branches of geometry and topology. In this article, we will talk about the characteristics, types of cylinders, and some formulas related to cylindrical structures.

Moreover, we have provided some solved examples of common mathematical formulas involving the area and volume of cylinders. The definitions and solved examples given here are meant to help class 8, 9 and 10 students understand cylindrical figures better. Read on to know more about cylindrical figures.

## Definition, Properties, and Formulas of Cylindrical Structures

A cylinder is a basic three-dimensional geometric object, with one curved surface known as the lateral surface and two circular surfaces at the ends. The cylinder has three faces, two edges (where two faces meet) and NO vertices (corners where two edges meet) as it has no corners.

### Cylinder Properties

A cylinder has some unique properties.

1. A cylindrical structures has a lateral surface and two bases. Total surface area is the sum of the area of the lateral surface and two bases.
2. The bases are parallel and identical.
3. It is like the prism since it has the same cross-section everywhere.
4. A cylinder can have two types of bases, elliptical and circular.

### Types of Cylinder

There are four types of cylindrical structures that can be found around.
Let us see how they look.

1. Right circular cylindrical structures
2. Oblique cylindrical structures
3. Elliptical cylindrical structures
4. Cylindrical shell or hollow cylindrical structures

#### Right Circular Cylinder

A right circular cylinder is an object formed by rolling the rectangle on one of its sides as an axis.
If the axis (one of the sides of the rectangle) is perpendicular to the radius $$\left( r \right),$$ then the cylinder is called a right circular cylinder.
The base and top of the cylinder are circular and they are parallel to each other, the distance between these circular faces of a cylinder is known as the height $$\left( h \right),$$ of a cylinder.

#### Oblique Cylinder

If the circular faces are not over each other but sideways, and the axis produces an angle other than the right angle to the bases, then it is called an oblique cylinder.

#### Elliptical Cylinder

A cylinder with elliptical shaped bases is known as an elliptical cylindrical structure.

#### Cylindrical Shell or Hollow Cylinder

A hollow cylinder is a cylindrical structure that is empty from the inside and has a difference in the outer and the inner surface radius of a cylinder. It can have different inner lateral surface areas and outer lateral surface areas as inner and outer radii are not the same.

Some Examples of Hollow Cylinder:

### Examples of Cylindrical Shaped Objects

Battery, iron rod, wooden stick for people, candles, water bottle, and gas cylinder are a few examples of cylindrical shape.

### What is the Area of Cylinder?

The space taken by a flat shape or the surface of an object is known as the area. The area of a figure is the number of unit squares that enfold the surface of a closed figure. We can measure the area in square units such as square centimetres, square inches, square feet, etc.

#### Formula of the Area of Cylinder

The area of the cylinder is the total space occupied by a cylinder in three-dimensional geometry. The area of a cylinder is equal to the sum of the lateral surface area and the area of two circular bases. There is a curved or lateral surface are in between two circular bases. When the curved surface opened, it stands for a rectangular figure. The different factors that are applied to derive the cylinder area are height, radius, axis, side and base. The radius of the cylinder is considered as the radius of two circular faces. The radius of the cylinder can be represented as $$\left(r\right)$$ and the perpendicular distance between two circular faces is known as the height of the cylinder. The height of the cylinder is defined as $$\left(h\right)$$

The surface area of a cylindrical structure can be classified into two parts.

1. Curved surface area (CSA)
2. Total surface area (TSA)

#### Curved Surface Area of Cylinder

Curved surface area is also known as lateral surface area. Curved surface area means the area of the cylindrical structure without the area of the circular base.
In general, the area is measured using square units such as centimetre square, meter square etc.
If the curved surface area of the cylinder is opened, then a rectangular shape is found.
The curved surface area includes two circular edges that can be the same as the circumference of the circle.
The formula of the circumference of the circle is $$2\pi r.$$
So, it can be said that the length of the rectangle (after opening the curved surface) is $$2\pi r$$ and breadth is $$\left( h \right),$$
Therefore, the area of the curved surface is $$2\pi r \times h = 2\pi rh$$ (as the area of a rectangle is length $$\times$$ breadth or $$l \times b$$).

#### Total Surface Area of Cylinder

Total surface area is the summation of curved surface area and the area of two circular bases.
Total surface area $$=$$ curved surface area $$+$$ area of top circular base + area of the circular bottom base.
Therefore, the total surface area $$= 2\pi r h + 2\pi {r^2} = 2\pi r \left({h + r} \right)$$
[ As the area of the circle is $$\pi {r^2}$$]

### What is the Volume of Cylinder?

Volume is the space occupied by the matter (solid, liquid, gas, or plasma) inside the three-dimensional object or the volume of a three-dimensional object is generally defined as the object’s capacity, which can hold the matter.

For example:

1. Sand filled into a cylindrical container.
2. Water filled in a cylindrical tank.
3. Gas filled in a cylindrical container.
4. In general, volume is measured using the unit cubic metre or cubic centimetre and for liquid we use litre.

#### Volume of Cylinder: Formula

A cylindrical structure is a three-dimensional solid. In general, the volume of a three-dimensional shape is a product of its area of base and height.
The volume of a cylindrical structure is equal to the product of the area of the circular base and the height of the cylinder.

The volume of a cylinder is measured in cubic units.

The volume of a cylinder $$=$$ Area of circle $$\times$$ Height
Area of circle $$= \pi {r^2}$$
The height of the right circular cylinder is $$(h).$$
The volume of a cylinder $$= \pi {r^2}h.$$

### Solved Examples

Question 1: The radius of a cylindrical structure is $$5\, {\text{cm}}$$ and the height is $$15\,{\text{cm}} {\text{.}}$$ What is the total surface area of the cylindrical structure? Use $$\pi = 3.14$$
Answer: Formula of the total surface area of a cylinder $$= 2\pi r \left({h + r} \right)$$ $${\text{square units}}$$. Therefore, area
$$= 2\pi r \left({h + r} \right) = 2\pi \times 5\left({15 + 5} \right) = 2 \times 3.14 \times 5 \times 20 = 628$$
Hence, the total surface area is $$628\, {\text{c}}{{\text{m}}^2}$$

Question 2: Find the lateral surface area when the radius of the cylinder is $$10\,{\text{cm}}$$ and height is $$20\,{\text{cm}}{\text{.}}$$ Use $$\pi = 3.14$$
Answer: The formula of lateral/curved surface area of a cylinder is $$2\pi rh.$$ Thus, $$2\pi rh = 2 \times 3.14 \times 10 \times 20\,{\text{c}}{{\text{m}}^2} = 1256\,{\text{c}}{{\text{m}}^2}$$
Hence, the lateral surface area is $$1256\,{\text{c}}{{\text{m}}^2}.$$

Question 3: Find the volume of a cylindrical shape oil container that has a height of $$8\,{\text{cm}}$$ and diameter of $$12\,{\text{cm?}}$$. Use $$\pi = 3.14$$
Answer: Given: Diameter of the container $$= 12\,{\text{cm}}$$
Thus, the radius of the container $$= \frac{{12}}{2}\,{\text{cm=6}}\,{\text{cm}}$$
Height of the container $${\text{=8}}\,{\text{cm}}$$
The formula of the volume of a cylindrical structure $${\text{=}}\pi {{\text{r}}^2}{\text{h}}$$ cubic units.
Therefore, the volume of the given container $$= \pi {\left( 6 \right)^2} \times 8\,{\text{c}}{{\text{m}}^3}$$
Volume $$= 3.14 \times \left( {{6^2}}\right) \times 8\,{\text{c}}{{\text{m}}^3} = 904.32\,{\text{c}}{{\text{m}}^3}$$
Hence, the volume is $$904.32\,{\text{c}}{{\text{m}}^3}.$$

Question 4: The volume of a cylindrical water tank is $$1000\,{{\text{m}}^3}$$ and the height of the tank is $$20\,{\text{m}}$$. Calculate the radius of the tank. Use $$\pi = 3.14$$
Answer: Given: Height $$\left( h \right)$$ of base $$= 20\,{\text{m}}$$ and volume of tank $$= 1000\,{{\text{m}}^3}$$
We know that volume of a cylindrical structure $$= \pi {r^2}h$$
So, $$1000 = 3.14 \times {r^2} \times 20$$
$$r = \sqrt {15.92\,} \, = 3.99 \approx 4\,{\text{m}}$$
Hence, the radius of the tank is $$4\,{\text{m}}$$ (approximately).

Question 5: Find the volume of a cylindrical metal pipe, whose length is $$40\,{\text{cm}}$$ and the outer radius is $$80\,{\text{cm,}}$$ and the thickness of the metal pipe is $$2\,{\text{cm}}{\text{.}}$$ Use $$\pi = 3.14$$
Length $$\left(h \right)$$ of the metal pipe $$= 40\,{\text{cm}}$$
The outer radius of the pipe $$= 80\,{\text{cm}}$$
The inner radius of the pipe $$=$$ Outer radius of the pipe $$-$$ Thickness of the metal pipe
The inner radius of the pipe $$= \left({80 – 2} \right)\, {\text{cm}} = 78\, {\text{cm}}$$
We know that volume of a hollow cylinder $$= \pi \left({{R^2} – {r^2}} \right)h$$
Now, the volume of a metal pipe $$= 3.14 \times \left({{ {80}^2} – {{78}^2}} \right) \times 40 = 39689\, {\text{c}}{{\text{m}}^3}.$$
Hence, the volume of a metal pipe is $$39689.6\,{\text{c}}{{\text{m}}^3}.$$

### Summary

A cylindrical structure has one curved surface and two bases, zero vertex, and two edges. It is a prism with two bases. Their shapes can be classified into four types; these are right circular cylinder, elliptical cylinder, oblique cylinder, and hollow cylindrical structures. Two bases of a cylindrical structure are parallel to each other and the difference between the two bases is known as altitude or height.