Azimjonov Alisherning Ingliz tili fanidan yozgan mustaqil ishi

Farg’ona davlat universiteti

Fizika-matematika fakulteti

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18.01 guruh talabasi

Azimjonov Alisherning

Ingliz tili fanidan yozgan

MUSTAQIL ISHI

Theme: Cylider

Cylinder, in geometry, surface of revolution that is traced by a straight line (the generatrix) that always moves parallel to itself or some fixed line or direction (the axis). The path, to be definite, is directed along a curve (the directrix), along which the line always glides. In a right circular cylinder, the directrix is a circle. The axis of this cylinder is a line through the centre of the circle, the line being perpendicular to the plane of the circle. In an oblique circular cylinder, the angle that the axis makes with the circle is other than 90°.

Cylinder

QUICK FACTS

RELATED TOPICS

Generatrix

Helix

The directrix of a cylinder need not be a circle, and if the cylinder is right, planes parallel to the plane of the directrix that intersect the cylinder produce intersections that take the shape of the directrix. For such a plane, if the directrix is an ellipse, the intersection is an ellipse.

The generatrix of a cylinder is assumed to be infinite in length; the cylinder so generated, therefore, extends infinitely in both directions of its axis. A finite cylinder has a finite base, the surface enclosed by the directrix, and a finite length of generatrix, called an element.

Cylinder is one of the basic 3d shapes, in geometry, which has two parallel circular bases at a distance. The two circular bases are joined by a curved surface, at a fixed distance from the center. The line segment joining the center of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas-cylinder is one of the real-life examples of cylinders.

Since, the cylinder is a three-dimensional shape, therefore it has two major properties, i.e., surface area and volume. The total surface area of the cylinder is equal to the sum of its curved surface area and area of the two circular bases. The space occupied by a cylinder in three dimensions is called its volume.

Here we will learn about its definition, formulas, properties of cylinder and will solve some examples based on them. Apart from this figure, we have concepts of Sphere, Cone, Cuboid, Cube, etc. which we learn in Solid Geometry.

Table of contents:

Definition

Shape of Cylinder

Properties

Formulas

Examples

FAQs

Definition

In mathematics, a cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. These bases are normally circular in shape (like a circle) and the center of the two bases are joined by a line segment, which is called the axis. The perpendicular distance between the bases is the height, “h” and the distance from the axis to the outer surface is the radius “r” of the cylinder.

Below is the figure of the cylinder showing area and height.

Cylinder

Cylinder Shape

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder. The line segment joining the two centers is the axis, that denotes the height of the cylinder.

The top view of the cylinder looks like a circle and the side view of the cylinder looks like a rectangle.

Unlike cones, cube and cuboid, a cylinder does not have any vertices, since the cylinder has a curved shape and no straight lines. It has two circular faces.

Properties

Each shape has some properties that differentiate one shape from another. Therefore, cylinders also have its characteristics.

The bases are always congruent and parallel.

If the axis forms a right angle with the bases, which are exactly over each other, then it is called a “Right Cylinder”.

It is similar to the prism since it has the same cross-section everywhere.

If the bases are not exactly over each other but sideways, and the axis does not produce the right angle to the bases, then it is called “Oblique Cylinder”.

If the bases are circular in shape, then it is called a right circular cylinder.

If the bases are in an elliptical shape, then it is called an “Elliptical Cylinder”.

To learn more properties of cylinder.

Formulas

The cylinder has three major quantities, based on which we have the formulas.

Lateral Surface Area or curved surface area

Total Surface Area

Volume

Let us see, their formulas now.

Curved Surface Area of Cylinder

The area of the curved surface of the cylinder which is contained between the two parallel circular bases. It is also stated as a lateral surface area. The formula for it is given by:

Curved Surface Area = 2πrh square units

Total Surface Area of Cylinder

The total surface area of a cylinder is the sum of curved surafce area and the area of two circular bases.

TSA = Curved surface + Area of Circular bases

TSA = 2πrh + 2πr2

We can see, from the above expression, 2πr is common. Therefore,

Total surface area, A = 2πr(r+h) square units

Volume of Cylinder

Every three dimensional shape or a solid has volume that occupies some space. The volume of the cylinder is the space occupied by it in any three-dimensional plane. The amount of water that could be immersed in a cylinder is described by its volume. The formula for the volume of cylinder is given by:

Volume of the Cylinder, V = πr2h cubic units

Where ‘h’ is the height and ‘r’ is the radius.

Also, read: Area Of Hollow Cylinder

Solved Examples

Question 1: Find the total surface area of the cylinder, whose radius is 5cm and height is 10cm?

Solution: We know, from the formula,

Total surface area of a cylinder, A = 2πr(r+h) square units

Therefore, A = 2π × 5(5 + 10) = 2π × 5(15) = 2π × 75 = 150 × 3.14 = 471 cm2

Question 2: What is the volume of a cylindrical shape water container, that has a height of 7cm and diameter of 10cm?

Solution: Given,

Diameter of the container = 10cm

Thus, the radius of the container = 10/2 = 5cm

Height of the container = 7cm

As we know, from the formula,

Volume of a cylinder = πr2h cubic units.

Therefore, volume of the given container, V = π × 52 × 7

V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25

V = 550 cm3

Frequently Asked Questions – FAQs

What are the characteristics of cylinder?

A cylinder has two parallel circular bases and a curved surface. The perpendicular distance between the two bases is the height of cylinder. The line segment joining the center of two bases is the axis of cylinder.

How many vertices does a cylinder have?

A cylinder has two faces that are circular in shape and parallel to each other. But it does not have any vertices, since it is a curved shape.

What are the real life examples of cylinder?

The real life examples of cylinder are: Gas cylinder, fire extinguisher, cans, pipes, etc.

What are the formulas of cylinder?

The formulas of cylinder are:

Total surface area = 2πr(r+h) square units

Volume of cylinder = πr2h cubic units

How many bases does the cylinder have?

A cylinder is having two circular bases.

How to find total surface area of cylinder?

First find the curved surface of cylinder which is equal to 2πrh, where r is the radius and h is the height of cylinder. Then add this curved surface area with the areas of two circular bases of cylinder to get the total surface area.

Total surface area = Curved Surface area + Two circular base areas