## What is a cylinder?

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

## Terminologies

- Axis- An axis of a cylinder is the line passing through the centers of the two circular bases.
- Height- The height of a cylinder is the perpendicular distance between the centers of the two circular bases.
- Radius- The radius of a cylinder is the distance between the axis and the outer surface of the cylinder.

## Types of a cylinder

There are four types of cylinders- right circular cylinder, oblique cylinder, elliptical cylinder, and right circular hollow cylinder.

- In a
**right circular cylinder**, the axis of a cylinder is perpendicular to both of its bases, that is, when a right angle is formed between the axis and both bases, the cylinder is a right circular cylinder. - In an
**oblique cylinder**, the axis of a cylinder is not perpendicular to both of its bases, that is, when a right angle is not formed between the axis and both bases, the cylinder is known as the oblique cylinder. The oblique cylinder is formed when the bases are not exactly over each other, but sideways. - In an
**elliptical cylinder**, the bases of the cylinder are elliptical. - A
**right circular hollow cylinder**is formed when two right circular cylinders are bounded into one another, with some space in between them. Each cylinder shares the same axis and whose annular bases are parallel to each other. The right circular hollow cylinder is also known as a cylindrical shell.

## Properties of a cylinder

- A cylinder has one curved surface and two identical flat faces.
- The two faces of a cylinder are always congruent and parallel to each other.
- The two faces of a cylinder are always identical. They can be either circular or elliptical.
- A cylinder does not have any vertex or corner.
- The size of a cylinder depends on the radius of the flat surfaces and the height of the curved surface.

## Volume and area of a cylinder

The calculations of a cylinder are primarily done for the three parameters- Volume, curved surface area, and total surface area.

### The volume of a cylinder

The volume of a cylinder is the total space occupied by the cylinder. The formula for the volume of a cylinder is derived using Cavalieri's principle.

Consider a right circular cylinder with the radius of base = $r$ and height = $h$. Hence, the volume of the cylinder can be given as,

$Volume=areaofcircularbase\times height\phantom{\rule{0ex}{0ex}}Volume=\pi {r}^{2}\times h\phantom{\rule{0ex}{0ex}}Volume={\mathrm{\pi r}}^{2}\mathrm{h}$

### The curved surface area of a cylinder

The curved surface area of a cylinder is the space occupied between the two parallel circular bases, that is, the area occupied by the curved surface of a cylinder. The curved surface area of a cylinder is also known as the lateral surface area of a cylinder.

Consider a right circular cylinder having the radius of base = $r$ and height = $h$. Hence, the curved surface area of a cylinder can be given as,

$Curvedsurfaceareaofacylinder=2\mathrm{\pi rh}$

### The total surface area of a cylinder

The total surface area of a cylinder is the total area occupied by the cylinder, that is, the area of the curved surface and the area of the bases. Hence, the total surface area of a cylinder will be equal to the sum of areas of the two bases and the area of the curved surface of the cylinder.

Consider a right circular cylinder having the radius of base = $r$ and height =$h$. Hence, the total surface area of a cylinder can be given as,

$Totalsurfaceareaofacylinder=Areaofbases+Curvedsurfacearea\phantom{\rule{0ex}{0ex}}TSA=\pi {r}^{2}+\pi {r}^{2}+2\mathrm{\pi r}h\phantom{\rule{0ex}{0ex}}TSA=2\pi {r}^{2}+2\mathrm{\pi r}h\phantom{\rule{0ex}{0ex}}TSA=2\pi r(r+h)$

## Steinmetz Solid

A Steinmetz solid is a geometric figure formed by intersecting two or three cylinders of equal radius at right angles. The intersection of two cylinders is known as bicylinder and the intersection of three cylinders is known as tricylinder.

Consider the cylinders of radius = $r$.

Hence, the formulas for bicylinder and tricylinder can be given as,

For bicylinder, the volume is given as $\frac{16}{3}{r}^{3}$ and the surface area is given as $16{r}^{2}$.

For tricylinder, the volume is given as $8(2-\sqrt{2}).{r}^{3}$ and the surface area is given as $24(2-\sqrt{2}).{r}^{2}$

The bicylinder and tricylinder are shown below-

## Formulas

$Volumeofacylinder={\pi}^{2}h$

$Curvedsurfaceareaofacylinder=2\pi rh$

$Totalsurfaceareaofacylinder=2\pi r(r+h)$

## Context and Applications

The cylinders are useful in the real life. Various objects that we see daily such as the LPG cylinders, cold-drink cans, water tanks, remote batteries, pipes, beakers, and so on are cylindrical. The cylinders are useful for students studying in schools, various undergraduate and postgraduate courses:

- Bachelors in Engineering (Mechanical and Civil)
- Masters in Engineering (Mechanical and Civil)
- Bachelors in Science (Mathematics)
- Masters in Science (Mathematics)

## Practice Problems

Q1. Which of the following is the line passing through the centers of the two circular bases of a cylinder?

- Base
- Height
- Axis
- Co-ordinate

Correct Answer: Option c

Explanation: An axis of a cylinder is the line passing through the centers of the two circular bases.

Q2. Which of the following is the perpendicular distance between the centers of the two circular bases of a cylinder?

- Base
- Height
- Radius
- Semi-axis

Correct Answer: Option b

Explanation: The height of a cylinder is the perpendicular distance between the centers of the two circular bases.

Q3. What is the cylinder with an elliptical base known as?

- Elliptical cylinder
- Right circular cylinder
- Oblique cylinder
- Right circular hollow cylinder

Correct Answer: Option a

Explanation: A cylinder whose bases are elliptical is known as an elliptical cylinder.

Q4. Which of the following is the area occupied by only the curved surface of a cylinder?

- The total surface area of the cylinder
- The partial surface area of a cylinder
- The half surface area of a cylinder
- The lateral surface area of a cylinder

Correct Answer: Option d

Explanation: The lateral surface area of a cylinder is the area occupied by only the curved surface of a cylinder.

Q5. Which of the following is not a property of a cylinder?

- A cylinder has one curved surface and two identical flat faces.
- The two faces of a cylinder are always congruent and parallel to each other.
- The cylinder has four vertices.
- The size of a cylinder depends on the radius of the flat surfaces and the height of the curved surface.

Correct Answer: Option c

Explanation: The cylinder has four vertices is not a property of a cylinder.

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