Definition of the base of the cylinder. Cylinder as a geometric figure

A cylindrical surface is formed by moving a straight line parallel to itself. The point of the line that is selected moves along the given plane curve - guide. This line is called generatrix of a cylindrical surface.

Direct cylinder- this is a cylinder in which the generators are perpendicular to the base. If the generators of the cylinder are not perpendicular to the base, then this will be inclined cylinder.

Circular cylinder- a cylinder whose base is a circle.

Round cylinder- a cylinder that is both straight and circular.

Straight circular cylinder determined by the radius of the base R and forming L, which is equal to the height of the cylinder H.

A prism is a special case of a cylinder.

Formulas for finding the elements of a cylinder.

Lateral surface area of ​​a right circular cylinder:

S side = 2πRH

Total surface area of ​​a right circular cylinder:

S=Sside+2Sbasic = 2 π R(H+R)

Volume of a straight circular cylinder:

V = S main H = πR 2 H

A straight circular cylinder with a beveled base or a briefly beveled cylinder is determined using the base radius R, minimum height h 1 and maximum height h 2.

Lateral surface area of ​​a beveled cylinder:

S side = πR(h 1 + h 2)

The area of ​​the base of a beveled cylinder.

Cylinder(ancient Greek κύλινδρος - roller, roller) - a geometric body bounded by a cylindrical surface and two parallel planes intersecting it. A cylindrical surface is a surface obtained by such a translational movement of a straight line (generator) in space that the selected point of the generatrix moves along a flat curve (director). The part of the cylinder surface limited by the cylindrical surface is called the lateral surface of the cylinder. The other part, bounded by parallel planes, is the base of the cylinder. Thus, the border of the base will coincide in shape with the guide.

In most cases, a cylinder means a straight circular cylinder, the guide of which is the circle and the bases are perpendicular to the generatrix. Such a cylinder has an axis of symmetry.

Other types of cylinder - (according to the inclination of the generatrix) oblique or inclined (if the generatrix does not touch the base at a right angle); (according to the shape of the base) elliptic, hyperbolic, parabolic.

A prism is also a type of cylinder - with a polygon-shaped base.

Cylinder surface area

Lateral surface area

To calculate the area of ​​the lateral surface of a cylinder

The area of ​​the lateral surface of the cylinder is equal to the length of the generatrix, multiplied by the perimeter of the section of the cylinder by a plane perpendicular to the generatrix.

The lateral surface area of ​​a straight cylinder is calculated from its development. The development of a cylinder is a rectangle with a height and length equal to the perimeter of the base. Therefore, the area of ​​the lateral surface of the cylinder is equal to the area of ​​its development and is calculated by the formula:

In particular, for a right circular cylinder:

For an inclined cylinder, the area of ​​the lateral surface is equal to the length of the generatrix multiplied by the perimeter of the section perpendicular to the generatrix:

Unfortunately, a simple formula expressing the area of ​​the lateral surface of an oblique cylinder through the parameters of the base and height, unlike the volume, does not exist.

Total surface area

The total surface area of ​​a cylinder is equal to the sum of the areas of its lateral surface and its bases.

For a straight circular cylinder:

Cylinder volume

For an inclined cylinder there are two formulas:

For a straight cylinder , and , and the volume is equal to:

For a circular cylinder:

Where d- base diameter.

Notes

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See what “Cylinder” is in other dictionaries:

- (lat. cylindrus) 1) a geometric body bounded at the ends by two circles, and at the sides by a plane enveloping these circles. 2) in watchmaking: a special kind of double wheel lever. 3) a hat shaped like a cylinder. Dictionary of foreign words,... ... Dictionary of foreign words of the Russian language

cylinder- a, m. cylindre m., German. Zylinder, lat. cylindrus gr. 1. A geometric body formed by the rotation of a rectangle around one of its sides. Cylinder volume. BAS 1. The thickness of a cylinder is equal to the area of ​​its base multiplied by its height. Dal... Historical Dictionary of Gallicisms of the Russian Language

Male, Greek straight stack, shaft; oblik, oblyak; a body bounded at the ends by two circles, and at the sides by a plane bent in circles. The thickness of a cylinder is equal to the area of ​​its base multiplied by its height, geom. Steam cylinder, freebie, pipe in which... ... Dahl's Explanatory Dictionary

Cylindrical surface, drum, shaft; cap, hat, roller, roll, mandrel, cylinder, point, drawstring, body, roller Dictionary of Russian synonyms. cylinder noun, number of synonyms: 22 atactosteles (2) ... Dictionary of synonyms

- (from the Greek kylindros) in elementary geometry, a geometric body formed by rotating a rectangle about one side: the volume of the cylinder is V=?r2h, and the area of ​​the lateral surface is S = 2?rh. The lateral surface of the cylinder is part of the cylindrical... ...

A hollow part with a cylindrical inner surface in which a piston moves. One of the main parts of piston machines and mechanisms... Big Encyclopedic Dictionary

Tall men's hat made of silk plush with small hard brim... Big Encyclopedic Dictionary

CYLINDER, a solid or surface formed by rotating a rectangle about one of its sides as an axis. The volume of a cylinder, if we denote its height as h and the radius of its base as r, is equal to pr2h, and the area of ​​the curved surface is 2prh... Scientific and technical encyclopedic dictionary

CYLINDER, cylinder, male (from Greek kylindros). 1. A geometric body formed by the rotation of a rectangle around one of its sides, called the axis, and having a circle (mat.) at its base. 2. Part of the machines (engines, pumps, compressors, etc.) in... ... Ushakov's Explanatory Dictionary

CYLINDER, huh, husband. 1. A geometric body formed by rotating a rectangle around one of its sides. 2. Columnar-shaped object, e.g. part of a piston machine. 3. A tall, hard hat of this shape with a small brim. Black c. | adj.... ... Ozhegov's Explanatory Dictionary

- (Steam cylinder) one of the main parts of piston machines. It is made in the form of a hollow round center in which the piston moves. The center of steam engines is usually equipped with a steam jacket to heat its walls in order to reduce steam condensation.... ... Marine Dictionary

The name of the science “geometry” is translated as “earth measurement”. It originated through the efforts of the very first ancient land managers. And it was like this: during the floods of the sacred Nile, streams of water sometimes washed away the boundaries of farmers’ plots, and the new boundaries might not coincide with the old ones. Taxes were paid by peasants to the treasury of the pharaoh in proportion to the size of the land allotment. Special people were involved in measuring the areas of arable land within the new boundaries after the spill. It was as a result of their activities that a new science arose, which was developed in Ancient Greece. There it received its name and acquired an almost modern appearance. Subsequently, the term became an international name for the science of flat and three-dimensional figures.

Planimetry is a branch of geometry dealing with the study of plane figures. Another branch of science is stereometry, which examines the properties of spatial (volumetric) figures. Such figures include the one described in this article - a cylinder.

There are plenty of examples of the presence of cylindrical objects in everyday life. Almost all rotating parts - shafts, bushings, journals, axles, etc. - have a cylindrical (much less often - conical) shape. The cylinder is also widely used in construction: towers, support columns, decorative columns. And also dishes, some types of packaging, pipes of various diameters. And finally - the famous hats, which have long become a symbol of male elegance. The list goes on and on.

Definition of a cylinder as a geometric figure

A cylinder (circular cylinder) is usually called a figure consisting of two circles, which, if desired, are combined using parallel translation. These circles are the bases of the cylinder. But the lines (straight segments) connecting the corresponding points are called “generators”.

It is important that the bases of the cylinder are always equal (if this condition is not met, then we have a truncated cone, something else, but not a cylinder) and are in parallel planes. The segments connecting the corresponding points on circles are parallel and equal.

The set of an infinite number of forming elements is nothing more than the lateral surface of the cylinder - one of the elements of a given geometric figure. Its other important component is the circles discussed above. They are called bases.

Types of cylinders

The simplest and most common type of cylinder is circular. It is formed by two regular circles acting as bases. But instead of them there may be other figures.

The bases of the cylinders can form (in addition to circles) ellipses and other closed figures. But the cylinder may not necessarily have a closed shape. For example, the base of a cylinder can be a parabola, a hyperbola, or another open function. Such a cylinder will be open or deployed.

According to the angle of inclination of the cylinders forming the bases, they can be straight or inclined. For a straight cylinder, the generatrices are strictly perpendicular to the plane of the base. If this angle is different from 90°, the cylinder is inclined.

What is a surface of revolution

The straight circular cylinder is without a doubt the most common rotating surface used in engineering. Sometimes, for technical reasons, conical, spherical, and some other types of surfaces are used, but 99% of all rotating shafts, axes, etc. are made in the form of cylinders. In order to better understand what a surface of revolution is, we can consider how the cylinder itself is formed.

Let's say there is a certain straight line a, located vertically. ABCD is a rectangle, one of whose sides (segment AB) lies on a line a. If we rotate a rectangle around a straight line, as shown in the figure, the volume that it will occupy while rotating will be our body of rotation - a right circular cylinder with height H = AB = DC and radius R = AD = BC.

In this case, as a result of rotating the figure - a rectangle - a cylinder is obtained. By rotating a triangle, you can get a cone, by rotating a semicircle - a ball, etc.

Cylinder surface area

In order to calculate the surface area of ​​​​an ordinary right circular cylinder, it is necessary to calculate the areas of the bases and lateral surfaces.

First, let's look at how the lateral surface area is calculated. This is the product of the circumference of the cylinder and the height of the cylinder. The circumference of a circle, in turn, is equal to twice the product of the universal number P by the radius of the circle.

The area of ​​a circle is known to be equal to the product P per square radius. So, by adding the formulas for the area of ​​determining the lateral surface with the double expression for the area of ​​the base (there are two of them) and making simple algebraic transformations, we obtain the final expression for determining the surface area of ​​the cylinder.

Determining the volume of a figure

The volume of a cylinder is determined according to the standard scheme: the surface area of ​​the base is multiplied by the height.

Thus, the final formula looks like this: the desired value is defined as the product of the height of the body by the universal number P and by the square of the radius of the base.

The resulting formula, it must be said, is applicable to solving the most unexpected problems. In the same way as the volume of the cylinder, for example, the volume of electrical wiring is determined. This may be necessary to calculate the mass of the wires.

The only difference in the formula is that instead of the radius of one cylinder there is the diameter of the wiring strand divided in half and the number of strands in the wire appears in the expression N. Also, instead of height, the length of the wire is used. In this way, the volume of the “cylinder” is calculated not just by one, but by the number of wires in the braid.

Such calculations are often required in practice. After all, a significant part of water containers are made in the form of a pipe. And it is often necessary to calculate the volume of a cylinder even in the household.

However, as already mentioned, the shape of the cylinder can be different. And in some cases it is necessary to calculate what the volume of an inclined cylinder is.

The difference is that the surface area of ​​the base is not multiplied by the length of the generatrix, as in the case of a straight cylinder, but by the distance between the planes - a perpendicular segment constructed between them.

As can be seen from the figure, such a segment is equal to the product of the length of the generatrix and the sine of the angle of inclination of the generatrix to the plane.

How to build a cylinder development

In some cases, it is necessary to cut out a cylinder ream. The figure below shows the rules by which a blank is constructed for the manufacture of a cylinder with a given height and diameter.

Please note that the drawing is shown without seams.

Differences between a beveled cylinder

Let us imagine a certain straight cylinder, bounded on one side by a plane perpendicular to the generators. But the plane bounding the cylinder on the other side is not perpendicular to the generators and not parallel to the first plane.

The figure shows a beveled cylinder. Plane A at a certain angle, different from 90° to the generators, intersects the figure.

This geometric shape is more often found in practice in the form of pipeline connections (elbows). But there are even buildings built in the form of a beveled cylinder.

Geometric characteristics of a beveled cylinder

The tilt of one of the planes of a beveled cylinder slightly changes the procedure for calculating both the surface area of ​​such a figure and its volume.

We’ll start a new topic online, and when I arrive we’ll conduct a test and test on the topic “Motion and Vectors.”

• We are starting to get acquainted with a new class of geometric bodies - bodies of revolution. The first representative of this class that we meet is the cylinder.
• Why is a cylinder called a body of revolution?

C cylinder, obtained by rotating a rectangle around one of its sides.

• The cylinder consists of two circles and many segments.
• Cylinderis a geometric body consisting of two equal circles located in parallel planes and a set of segments connecting the corresponding points of these circles.
• Cylinder Element Definitions:

Cylinder bases– equal circles located in parallel planes

Cylinder height- This the distance between the planes of its bases.

Cylinder axis- this is a straight line passing through the centers of the base of the cylinder (the axis of the cylinder is the axis of rotation of the cylinder).

Axial section of the cylinder– section of the cylinder by a plane passing through the axis of the cylinder (the axial section of the cylinder is the plane of symmetry of the cylinder). All axial sections of the cylinder are equal rectangles

Generator of the cylinder- this is a segment connecting a point on the circle of the upper base with a corresponding point on the circle of the lower base. All generatrices are parallel to the axis of rotation and have the same length, equal to the height of the cylinder.

When rotating around an axis, the generatrix of the cylinder formslateral (cylindrical) surface of the cylinder.

Cylinder radiusis the radius of its base.

Straight cylinder- This is a cylinder, the generatrices of which are perpendicular to the base.

Equal-area cylinder– a cylinder whose height is equal to its diameter (show an equal cylinder: using the button with the hand icon, switch the model back to interactive mode and change the height and radius values ​​of the proposed model so that ).

• Derivation of the formula for lateral surface area.

  The development of the lateral surface of the cylinder is a rectangle with sidesH And C, Where His the height of the cylinder, andC– length of the base circumference. Let us obtain formulas for calculating the areas of the lateralS b and full S n surfaces: S b = H · C= 2π RH, S n = S b + 2 S= 2π R(R + H).

Consolidation

Task No. 1. Calculate the lateral and total surface area of ​​a cylinder whose radius is 3 cm and height 5 cm (pi and round the answer to whole numbers).

2. The height of the cylinder ish, base radiusR. Find the cross-sectional area of ​​a plane drawn parallel to the cylinder axis at a distancea from her.

Homework: 522, 524, 526.

• R.S/ If anyone is interested, try following the link and looking at the electronic resource about the cylinder. First, on the page, install the OMS module on your PC and download the module. On the table that pops up, click play. And then look through all the pages in order.
• THANK YOU ALL.

Cylinder (circular cylinder)- a body that consists of two circles that do not lie in the same plane and are combined by parallel translation, and all segments connecting the corresponding points of these circles.

The circles are called cylinder bases , and the segments connecting the corresponding points of the circles’ circumferences are forming the cylinder. These segments form a cylindrical surface, which is lateral surface of the cylinder .

If the bases of the cylinder are not circles, then the cylinder can be elliptical. Typically, such types of cylinders are not considered in elementary geometry.

Alternative definition.

A cylinder is a geometric body bounded by a cylindrical surface and two parallel planes intersecting it.

Full surface The cylinder consists of a base and a side surface.

The cylinder is called direct, if its generators are perpendicular to the plane of the bases.

Radius The radius of a cylinder is called the radius of its base.

Height of a cylinder is the distance between the planes of its bases.

Axis of a cylinder is called a straight line passing through the centers of the base. It is parallel to the generators.

The cross section of a cylinder with a plane parallel to its axis is a rectangle. Its two sides are the generators of the cylinder, and the other two are parallel chords of the bases. The axial section of a cylinder is a section by a plane passing through its axis.

Cylinder volume

The volume of a cylinder is equal to the product of the area of ​​the base and the height N :

If only the area of ​​the base and the generatrix are known for a cylinder, then the volume of such a cylinder will be equal to the product of the area of ​​the base and the generatrix and the sine of the angle between the base and the generatrix.

For a cylinder with a circle at its base, the volume of the cylinder will be equal to the area of ​​the circle times the height.

Cylinder side surface area