Lesson topic: Cylinder, its elements.
The purpose of the lesson:
Consolidation of students' knowledge about the body of revolution - the cylinder (cylinder elements, formulas for the area of the lateral and full surface of the cylinder).
Student goal: be able to solve typical tasks for a cylinder in UNT tasks.
Lesson objectives:
1. to form skills for solving typical problems;
2. develop spatial representations on the example of round bodies;
3. continue the formation of logical and graphic skills.
Lesson type: combined.
Teaching methods: verbal, practical activity, work with a book, problematic.
Equipment: board, table number 3, a set of models.
During the classes
1. Organizational moment:
1. goal setting
2. psychological attitude.
2. Actualization of basic knowledge.
1) Work on cards.
Students are asked to complete a worksheet.
It is possible to work using copying (in this case, one copy is handed over to the teacher, and the second student checks in the course of further work in the lesson).
Card.
1. Draw the main elements of the cylinder on the drawing.
2
.Depict a) the axial section of the cylinder; b) section of the cylinder by a plane passing perpendicular to the axis of the cylinder; c) section of the cylinder by a plane parallel to the axis of the cylinder. What figure is obtained in each case?
3. Write down the formulas for calculating the surface area of a cylinder.
What can be found by these formulas? What should be known in these cases?
Students hand in worksheets.
3. Oral work on models. (in order to generalize knowledge and check the work done)
1) What shape is called a cylinder?
Cylinder - This is a geometric body consisting of two equal circles located in parallel planes and a set of segments connecting the corresponding points of these circles.
2) Why is a cylinder called a body of revolution?
A cylinder can be obtained by rotating a rectangle around one of its sides.
3) What are the types of cylinders?
Inclined cylinders, straight cylinders, cylindrical surfaces.
4) Name the elements of the cylinder.
Cylinder bases - equal circles located in parallel planes.
Cylinder height - This the distance between the planes of its bases.
Cylinder radius is the radius of its base.
Cylinder axis 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).
Cylinder generatrix - this is a segment connecting the point of the circle of the upper base with the corresponding point of the circle of the lower base. All generators are parallel to the axis of rotation and have the same length, equal to the height of the cylinder.
The generatrix of the cylinder during rotation around the axis forms lateral (cylindrical) surface of a cylinder .
5) What is a cylinder sweep?
The development of the lateral surface of the cylinder is a rectangle with sides H and C, where H is the height of the cylinder, and C is the circumference of the base.6) How to find the lateral surface area of a cylinder?
S b = H · C = 2 π RH
7) How to find the total surface area of a cylinder?
S P = S b + 2 S = 2 π R (R + H ).
8) What are the main types of sections of the cylinder. What figure is obtained in each case?
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.
cross section plane parallel to the axis of the cylinder. The section is a rectangle.
Plane section perpendicular to the axis of the cylinder. Circles in cross section, equal to the base.
9) Give examples of the use of cylinders.
Cylindrical gastronomy. Cylindrical architecture. Cylinders of the pharaoh (student performance 1-2 minutes).
4. Fixing the material. Problem solving.
At 
Students see a list of tasks for classwork. If desired, students have the opportunity to decide ahead of the mark.
№ 2 . The axial section of the cylinder is a square whose diagonal is 20 cm. Find: a) the height of the cylinder; b) So cylinder.
№3 The area of the axial section of the cylinder is 10 m 2, and the base area is 5 m 2. Find the height of the cylinder.
№4 The ends of the segment AB lie on different bases of the cylinder. The radius of the cylinder is r, his high - h, the distance between the straight line AB and the axis of the cylinder is d. Find: a) height if r = 10, d= 8, AB = 13.
№5* Two secant planes are drawn through the generatrix AA 1 of the cylinder, one of which passes through the axis of the cylinder. Find the ratio of the cross-sectional areas of the cylinder by these planes if the angle between them is equal to j.
5. Educational independent work. Independent work on options. (It is possible to organize pair work).
Plane g, parallel to the axis of the cylinder, cuts off the arc A from the circumference of the base m D with degree measure a . The radius of the cylinder is a, the height is h, the distance between the axis of the cylinder OO 1 and the plane g is equal to d.
Option 2. 1) Make a plan for calculating the cross-sectional area from the data a , h, d.2) Find AD if a =8 cm, a = 120°. 6. Setting homework . Repeat formula 1 and solve number 25. 7. Reflective-evaluative block.Reflection. What new did you learn in the lesson?
What have you learned?
What is your mood at the end of the lesson?
Can you explain the solution to these problems to a classmate who missed class today?
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 height 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 a 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:
Wikimedia Foundation. 2010 .
See what "Cylinder (geometry)" is in other dictionaries:
A branch of mathematics that studies the properties of various shapes (points, lines, angles, two-dimensional and three-dimensional objects), their size and relative position. For the convenience of teaching, geometry is divided into planimetry and solid geometry. AT… … Collier Encyclopedia
- (γήμετρώ earth, μετρώ measure). The concepts of space, position and form are among the original, with which man was already familiar in ancient times. The first steps in Georgia were made by the Egyptians and the Chaldeans. In Greece, G. was introduced ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
FREE SURFACE GEOMETRY- the shape of the free surface, formed under the action of gravity and centrifugal force during the rotation of the liquid metal around the axis of rotation. With a horizontal axis of rotation, the free surface is a circular cylinder, with a vertical ... Metallurgical Dictionary
A section of geometry in which geometric images are studied by methods of mathematical analysis. The main objects of DG are arbitrary sufficiently smooth curves (lines) and surfaces of Euclidean space, as well as families of lines and ...
This term has other meanings, see Pyramidatsu (meanings). The reliability of this section of the article has been questioned. It is necessary to verify the accuracy of the facts stated in this section. There may be explanations on the talk page ... Wikipedia
A theory that studies external geometry and the relationship between external and internal. geometry of submanifolds of Euclidean or Riemannian space. P. m. g. is a generalization of the classic. differential geometry of surfaces in Euclidean space. ... ... Mathematical Encyclopedia
Cartesian coordinate system Analytic geometry section of geometry in which ... Wikipedia
Section of geometry, in which geometrical are studied. images, primarily curves and surfaces, by mathematical methods. analysis. Usually in DGs the properties of curves and surfaces are studied in the small, that is, the properties of arbitrarily small pieces of them. Besides, in … Mathematical Encyclopedia
This term has other meanings, see Scope (meanings). Volume is an additive function of a set (measure) that characterizes the capacity of a region of space that it occupies. Initially, it arose and was applied without strict ... ... Wikipedia
A part of geometry included in elementary mathematics (See elementary mathematics). The boundaries of egalitarianism, as well as of elementary mathematics in general, are not strictly delineated. They say that E. g. is that part of geometry that is studied in ... ... Great Soviet Encyclopedia
Books
- Cheerful geometry for the little ones, Timofeevsky Alexander Pavlovich. A new book by the wonderful poet, author of the well-known Song of the Crocodile Gena Alexander Timofeevsky with vivid illustrations by Leonid Shmelkov in a playful way introduces kids to the main ...
Cylinder (straight circular cylinder) a body is called a body consisting of two circles (cylinder bases) combined by parallel translation, and all segments connecting the corresponding points of these circles during parallel translation. The segments connecting the corresponding points of the circles of the bases are called the generators of the cylinder.
Here is another definition:
Cylinder- a body that is bounded by a cylindrical surface with a closed guide and two parallel planes intersecting the generators of this surface.
Cylindrical surface- a surface that is formed by the movement of a straight line along a certain curve. The straight line is called the generatrix of the cylindrical surface, and the curved line is called the guide of the cylindrical surface.
Lateral surface of the cylinder- part of a cylindrical surface that is bounded by parallel planes.
Cylinder bases- parts of parallel planes cut off by the side surface of the cylinder.
Fig.1 mini
The cylinder is called direct(Cm. Fig.1) if its generators are perpendicular to the planes of the bases. Otherwise, the cylinder is called oblique.
circular cylinder- a cylinder whose bases are circles.
Right circular cylinder (just a cylinder) is the body obtained by rotating a rectangle around one of its sides. Cm. Fig.1.
Cylinder radius is the radius of its base.
Cylinder generatrix- generatrix of a cylindrical surface.
cylinder height called the distance between the planes of the bases. Cylinder axis is called a straight line passing through the centers of the bases. The section of a cylinder by a plane passing through the axis of the cylinder is called axial section.
The axis of the cylinder is parallel to its generatrix and is the axis of symmetry of the cylinder.
The plane passing through the generatrix of a right cylinder and perpendicular to the axial section drawn through this generatrix is called tangent plane of the cylinder. Cm. Fig.2.
Reaming of the side surface of the cylinder- a rectangle with sides equal to the height of the cylinder and the circumference of the base.
Cylinder side surface area- the area of the development of the lateral surface. $$S_(side)=2\pi\cdot rh$$ , where h is the height of the cylinder, and r is the radius of the base.
Full surface area of a cylinder- area, which is equal to the sum of the areas of the two bases of the cylinder and its lateral surface, i.e. is expressed by the formula: $$S_(full)=2\pi\cdot r^2 + 2\pi\cdot rh = 2\pi\cdot r(r+h)$$ , where h is the height of the cylinder, and r is the radius of the base.
Volume of any cylinder is equal to the product of the area of the base and the height: $$V = S\cdot h$$ Volume of a round cylinder: $$V=\pi r^2 \cdot h$$ , where ( r is the radius of the base).
A prism is a particular form of a cylinder (generators are parallel to the side edges; the guide is a polygon lying at the base). On the other hand, an arbitrary cylinder can be viewed as a degenerate ("smoothed") prism with a very large number of very narrow faces. Practically the cylinder is indistinguishable from such a prism. All properties of the prism are preserved in the cylinder.
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 a new science arose, which was developed in ancient Greece. There it received its name, and acquired an almost 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 of plane 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.
There are plenty of examples of the presence of cylindrical objects in everyday life. 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 this angle differs 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.
In this case, as a result of the rotation of the 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 base area (there are two of them) and performing 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.
Thus, the final formula looks like this: 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.
This geometric shape is 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.
Body of rotation called a body formed as a result of the rotation of a line around a straight line.
CYLINDER
A cylinder (circular cylinder) is 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 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.
Since parallel translation is motion, the bases of the cylinder are equal. Since during parallel transfer the plane passes into a parallel plane, then the bases of the cylinder lie in parallel planes. Since, during parallel translation, the points are displaced along parallel lines by the same distance, the generators of the cylinder are parallel and equal. The surface of a cylinder consists of bases and a side surface.
The radius of a cylinder is the radius of its base. The height of a cylinder is the distance between the planes of its bases. The axis of a cylinder is a straight line passing through the centers of the bases.
A cylinder is called straight if its generators are perpendicular to the planes of the bases. We will consider only a right circular cylinder, calling it simply a cylinder for brevity.
A cylinder can be obtained by rotating a rectangle around one of its sides. The figure shows a cylinder obtained by rotating the rectangle ABCD around the side AB. In this case, the side surface of the cylinder is formed by rotating the side CD, and the base - by rotating the sides BC and AD.
Cylinder sections
1) If the cutting plane passes through the axis of the cylinder, then the section is a rectangle (see figure), two sides of which are generatrices, and the other two are the diameters of the bases of the cylinder. Such a section is called axial.
