What is a circular cylinder. Reference abstract on geometry on the topic "cylinder"
Cylinder(other Greek. κύλινδρος - roller, skating rink) - geometric bodylimited by a cylindrical surface and two parallel planes intersecting it. Cylindrical surface - the surface obtained in such a way progressive movement straight line (generator) in space, that the selected point of the generatrix moves along a flat curve (guide). The part of the surface of the cylinder bounded 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, in which the guide is a circle and the bases are perpendicular to the generatrix. Such a cylinder has an axis of symmetry.
Other types of cylinder - (according to the slope 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) elliptical, hyperbolic, parabolic.
A prism is also a kind of cylinder - with a base in the form of a polygon.
Cylinder surface area
Lateral surface area
To the calculation of the lateral surface area 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 the cylinder is a rectangle with 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:
, andFor an inclined cylinder, the lateral surface area is equal to the length of the generatrix multiplied by the perimeter of the section perpendicular to the generatrix:
Unfortunately, there is no simple formula expressing the lateral surface area of an oblique cylinder in terms of base parameters and height, unlike volume.
Total surface area
Square full surface A cylinder is equal to the sum of the areas of its lateral surface and its bases.
For a straight circular cylinder:
Cylinder volume
There are two formulas for an inclined cylinder:
where is the length of the generatrix, and is the angle between the generatrix and the plane of the base. For a straight cylinderFor a straight cylinder , and , and the volume is:
For a circular cylinder:
where d- base diameter.
Notes
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Synonyms:See what "Cylinder" is in other dictionaries:
- (Latin cylindrus) 1) a geometric body bounded at the ends by two circles, from 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 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. ALS 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
Husband, Greek straight stack, shaft; oblets, 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. A steam cylinder, a freebie, a pipe in which ... ... Dictionary Dalia- a tall man's hat made of silk plush with small hard brim... Big encyclopedic Dictionary
CYLINDER, solid or a surface formed by revolving a rectangle about one of its sides as an axis. The volume of the cylinder, if we denote its height as h, and the radius of the base as r, is equal to pr2h, and the area of the curved surface is 2prh ... Scientific and technical encyclopedic dictionary
CYLINDER, cylinder, man. (from Greek kylindros). 1. A geometric body formed by the rotation of a rectangle about one of its sides, called the axis, and having a circle at the bases (mat.). 2. Part of the machines (motors, pumps, compressors, etc.) in ... ... Explanatory Dictionary of Ushakov
CYLINDER, a, husband. 1. A geometric body formed by the rotation of a rectangle around one of its sides. 2. Columnar object, eg. piston machine part. 3. A tall hard hat of this shape with a small brim. Black c. | adj… … Explanatory dictionary of Ozhegov
- (Steam cylinder) one of the main parts of piston machines. It is made in the form of a hollow round ring, in which the piston moves. C. steam engines 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 "measurement of the earth." It was born through the efforts of the very first ancient land surveyors. And it happened like this: during the floods of the sacred Nile, streams of water sometimes washed away the boundaries of the plots of farmers, and the new boundaries might not coincide with the old ones. Taxes were paid by the peasants to the treasury of the pharaoh in proportion to the size of the land allotment. After the spill, special people were engaged in measuring the areas of arable land within the new boundaries. It was as a result of their activities that the new science, developed in Ancient Greece. There she received the name, and acquired practically modern look. In the future, the term became the international name for the science of flat and three-dimensional figures.
Planimetry is a branch of geometry that deals with the study flat figures. Another branch of science is stereometry, which considers the properties of spatial (volumetric) figures. The cylinder described in this article also belongs to such figures.
Examples of the presence of cylindrical objects in Everyday life enough. Almost all parts of rotation - shafts, bushings, necks, axles, etc. have a cylindrical (much less often - conical) shape. The cylinder is widely used in construction: towers, supporting, decorative columns. And besides, dishes, some types of packaging, pipes of various diameters. And finally - the famous hats, which have become a symbol of male elegance for a long time. The list is endless.
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. It is these circles that 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 in front of us, something else, but not a cylinder) and are in parallel planes. The segments connecting the corresponding points on the circles are parallel and equal.
The totality of an infinite set of generators is nothing more than the lateral surface of a 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 (except for circles) ellipses and other closed figures. But the cylinder may not necessarily have a closed shape. For example, a parabola, a hyperbola, or another open function can serve as the base of a cylinder. Such a cylinder will be open or deployed.
According to the angle of inclination of the generatrices to the bases, the cylinders can be straight or inclined. For a right cylinder, the generators are strictly perpendicular to the plane of the base. If a given angle different from 90°, the cylinder is inclined.
What is a surface of revolution
A right circular cylinder is without a doubt the most common surface of revolution used in engineering. Sometimes, according to technical indications, conical, spherical, and some other types of surfaces are used, but 99% of all rotating shafts, axles, etc. 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 line a placed vertically. ABCD is a rectangle, one of whose sides (segment AB) lies on a straight 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 revolution - a right circular cylinder with height H = AB = DC and radius R = AD = BC.
AT this case, as a result of rotation of a figure - a rectangle - a cylinder is obtained. Rotating a triangle, you can get a cone, rotating a semicircle - a ball, etc.
Cylinder surface area
In order to calculate the surface area of an ordinary straight circular cylinder, it is necessary to calculate the areas of the bases and the lateral surface.
First, let's look at how the lateral surface area is calculated. This is the product of the circumference and the height of the cylinder. The circumference, in turn, is equal to twice the product of the universal number P to the radius of the circle.
The area of a circle is known to be equal to the product P to the square of the radius. So, adding the formulas for the area of determining the lateral surface with twice the 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 by the standard scheme: the surface area of the base is multiplied by the height.
So the final formula looks like in the following way: the desired is defined as the product of the height of the body by the universal number P and the square of the base radius.
The resulting formula, it must be said, is applicable to solving the most unexpected problems. In the same way as the volume of a cylinder, for example, the volume of electrical wiring is determined. This may be necessary to calculate the mass of wires.
The only difference in the formula is that instead of the radius of one cylinder, there is the diameter of the wiring core divided in two and the number of cores in the wire appears in the expression N. Also, wire length is used instead of height. Thus, the volume of the “cylinder” is calculated not by one, but by the number of wires in the braid.
Such calculations are often required in practice. After all, a significant part of the water tanks is 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 required to calculate what the volume of the inclined cylinder is equal to.
The difference is that the surface area of the base is multiplied not by the length of the generatrix, as in the case of a straight cylinder, but by the distance between the planes - a perpendicular segment built between them.
As can be seen from the figure, such a segment is equal to the product of the length of the generatrix by the sine of the angle of inclination of the generatrix to the plane.
How to build a cylinder sweep
In some cases, it is required to cut out a cylinder reamer. The figure below shows the rules by which a blank is built for the manufacture of a cylinder with a given height and diameter.
Please note that the figure is shown without seams.
Beveled Cylinder Differences
Let us imagine a 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 is not parallel to the first plane.
The figure shows a beveled cylinder. Plane a at some angle other than 90° to the generators, intersects the figure.
Such geometric shape more common 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 the beveled cylinder
The slope of one of the planes of the beveled cylinder slightly changes the order of calculation of both the surface area of such a figure and its volume.
Cylinder
Def. A cylinder is a body that consists of two circles aligned
parallel translation and all segments connecting the corresponding points
these circles.
The circles are called the bases of the cylinder, and the segments connecting the corresponding points of the circles of these circles are called the generators of the cylinder (Fig. 1)
rice. 1 fig. 2 fig. 3 fig. four
Cylinder properties:
1) The bases of the cylinder are equal and lie in parallel planes.
2) The generators of the cylinder are equal and parallel.
Def. The radius of a cylinder is the radius of its base.
Def. The height of a cylinder is the distance between the planes of its bases.
Def. The section of a cylinder by a plane passing through the axis of the cylinder is called an axial section.
The axial section of the cylinder is a rectangle with sides 2R and l(in a straight cylinder l= H) fig. 2
The cross section of the cylinder, parallel to its axis, are rectangles (Fig. 3).
The cross section of a cylinder with a plane parallel to the bases is a circle, equal to the bases(Fig. 4)
The surface area of a cylinder.
The lateral surface of the cylinder is composed of generators.
The full surface of a cylinder consists of the bases and the lateral surface.
S full = 2 S main + S side ; S main = P ∙ R 2 ; S side = 2 P ∙ R ∙NS full = 2PR ∙(R + H)
Practical part:
№1. The radius of the cylinder is 3cm and its height is 5cm. Find the area of the axial section and the area of \u200b\u200bthe half-
surface of the cylinder.
№2.
The diagonal of the axial section of the cylinder is inclined to the plane of the base at an angle 
and is equal to 20 cm. Find the area of the lateral surface of the cylinder.
№3. The radius of the cylinder is 2cm and its height is 3cm. Find the diagonal of the axial section of the cylinder.
№4.
The diagonal of the axial section of the cylinder, equal to 
, forms an angle with the plane of the base 
. Find the lateral surface area of the cylinder.
№5.
The lateral surface area of the cylinder is 15 
. Find the area of the axial section.
№6.
Find the height of the cylinder if its base area is 1 and S side = 
.
№7.
The diagonal of the axial section of the cylinder has a length of 8 cm and is inclined to the plane of the base at an angle 
. Find the total area of the cylinder.
A cylindrical chimney with a diameter of 65cm has a height of 18m. How much tin is needed to make it if 10% of the material is spent on the rivet?
Cylinder (circular cylinder) - a body that consists of two circles, combined by parallel transfer, and all segments connecting the corresponding points of these circles. The circles are called the bases of the cylinder, and the segments connecting the corresponding points of the circles of the circles are called the generators of the cylinder.
The bases of the cylinder are equal and lie in parallel planes, and the generators of the cylinder are parallel and equal. The surface of a cylinder consists of bases and a side surface. The lateral surface is formed by generators.
A cylinder is called straight if its generators are perpendicular to the planes of the base. A cylinder can be considered as a body obtained by rotating a rectangle around one of its sides as an axis. There are other types of cylinder - elliptical, hyperbolic, parabolic. A prism is also considered as a kind of cylinder.
Figure 2 shows an inclined cylinder. Circles with centers O and O 1 are its bases.
The radius of a cylinder is the radius of its base. The height of the cylinder is the distance between the planes of the bases. The axis of a cylinder is a straight line passing through the centers of the bases. It is parallel to the generators. The section of a cylinder by a plane passing through the axis of the cylinder is called an axial section. The plane passing through the generatrix of a straight cylinder and perpendicular to the axial section drawn through this generatrix is called the tangent plane of the cylinder.
A plane perpendicular to the axis of the cylinder intersects it side surface around the circumference equal circle grounds.
A prism inscribed in a cylinder is a prism whose bases are equal polygons inscribed in the bases of the cylinder. Its lateral edges are generatrices of the cylinder. A prism is said to be circumscribed near a cylinder if its bases are equal polygons circumscribed near the bases of the cylinder. The planes of its faces touch the side surface of the cylinder.
The area of the lateral surface of the cylinder can be calculated by multiplying the length of the generatrix by the perimeter of the section of the cylinder by a plane perpendicular to the generatrix.
The lateral surface area of a right cylinder can be found from its development. The development of the cylinder is a rectangle with height h and length P, which is 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:
P = 2πR, and Sb = 2πRh.
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:
S p = 2πRh + 2πR 2 = 2πR(h + R)
There are two formulas for finding the volume of an inclined cylinder.
You can find the volume by multiplying the length of the generatrix by the cross-sectional area of \u200b\u200bthe cylinder by a plane perpendicular to the generatrix.
The volume of an inclined cylinder is equal to the product of the area of the base and the height (the distance between the planes in which the bases lie):
V = Sh = S l sin α,
where l is the length of the generatrix, and α is the angle between the generatrix and the plane of the base. For a straight cylinder h = l.
The formula for finding the volume of a circular cylinder is as follows:
V \u003d π R 2 h \u003d π (d 2 / 4) h,
where d is the base diameter.
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kylindros, roller, roller) - a geometric body bounded by a cylindrical surface (called the side surface of the cylinder) and no more than two surfaces (cylinder bases); moreover, if there are two bases, then one is obtained from the other by parallel transfer along the generatrix of the lateral surface of the cylinder; and the base intersects each generatrix of the lateral surface exactly once.An infinite body bounded by a closed infinite cylindrical surface is called endless cylinder, bounded by a closed cylindrical ray and its base, is called open cylinder. The base and generators of a cylindrical beam are called the base and generators of an open cylinder, respectively.
A finite body bounded by a closed finite cylindrical surface and two sections that separate it is called final cylinder, or actually cylinder. The sections are called the bases of the cylinder. By definition of a finite cylindrical surface, the bases of a cylinder are equal.
Obviously, the generators of the lateral surface of the cylinder are equal in length (called tall cylinder) segments lying on parallel lines, and with their ends lying on the bases of the cylinder. Mathematical curiosities include the definition of any finite three-dimensional surface without self-intersections as a zero-height cylinder (this surface is considered simultaneously by both bases of the finite cylinder). The bases of the cylinder qualitatively affect the cylinder.
If the bases of the cylinder are flat (and hence the planes containing them are parallel), then the cylinder is called standing on the plane. If the bases of a cylinder standing on a plane are perpendicular to the generatrix, then the cylinder is called straight.
In particular, if the base of a cylinder standing on a plane is a circle, then one speaks of a circular (round) cylinder; if an ellipse - then elliptical.
The volume of the final cylinder is equal to the integral of the base area along the generatrix. In particular, the volume of a right circular cylinder is
(where is the radius of the base, is the height).
The lateral surface area of a cylinder is calculated using the following formula:
The total surface area of a cylinder is the sum of the lateral surface area and the area of the bases. For a straight circular cylinder:
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See what "Cylinder (geometry)" is in other dictionaries:
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