Abstract of the lesson on geometry "cylinder, its elements". Geometry reference abstract on the topic "cylinder" Irregular cylinder

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. 4

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).

Section of a cylinder by a plane parallel to the bases - 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?

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?

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.

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.

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?

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 its lateral surface along a circle equal to the circumference of the base.

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

Straight cylinder 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 generating L, which is equal to the height of the cylinder H.

A prism is a special case of a cylinder.

Formulas for finding 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+ 2Smain = 2 π R(H+R)

Volume of a straight circular cylinder:

V = S main H = πR 2 H

A straight circular cylinder with a chamfered base or a briefly chamfered cylinder is defined by the radius of the base R, minimum height h1 and maximum height h2.

Side surface area of ​​the bevelled cylinder:

S side \u003d πR (h 1 + h 2)

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

A cylinder is a geometric body bounded by two parallel planes and a cylindrical surface. In the article, we will talk about how to find the area of ​​a cylinder and, using the formula, we will solve several problems for example.

A cylinder has three surfaces: a top, a bottom, and a side surface.

The top and bottom of the cylinder are circles and are easy to identify.

It is known that the area of ​​a circle is equal to πr 2 . Therefore, the formula for the area of ​​two circles (top and bottom of the cylinder) will look like πr 2 + πr 2 = 2πr 2 .

The third, side surface of the cylinder, is the curved wall of the cylinder. In order to better represent this surface, let's try to transform it to get a recognizable shape. Imagine that a cylinder is an ordinary tin can that does not have a top lid and bottom. Let's make a vertical incision on the side wall from the top to the bottom of the jar (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).

After the full disclosure of the resulting jar, we will see a familiar figure (Step 3), this is a rectangle. The area of ​​a rectangle is easy to calculate. But before that, let us return for a moment to the original cylinder. The vertex of the original cylinder is a circle, and we know that the circumference of a circle is calculated by the formula: L = 2πr. It is marked in red in the figure.

When the side wall of the cylinder is fully expanded, we see that the circumference becomes the length of the resulting rectangle. The sides of this rectangle will be the circumference (L = 2πr) and the height of the cylinder (h). The area of ​​a rectangle is equal to the product of its sides - S = length x width = L x h = 2πr x h = 2πrh. As a result, we have obtained a formula for calculating the lateral surface area of ​​a cylinder.

The formula for the area of ​​the lateral surface of a cylinder
S side = 2prh

Full surface area of ​​a cylinder

Finally, if we add up the area of ​​all three surfaces, we get the formula for the total surface area of ​​a cylinder. The surface area of ​​the cylinder is equal to the area of ​​the top of the cylinder + the area of ​​the base of the cylinder + the area of ​​the side surface of the cylinder or S = πr 2 + πr 2 + 2πrh = 2πr 2 + 2πrh. Sometimes this expression is written by the identical formula 2πr (r + h).

The formula for the total surface area of ​​a cylinder
S = 2πr 2 + 2πrh = 2πr(r + h)
r is the radius of the cylinder, h is the height of the cylinder

Examples of calculating the surface area of ​​a cylinder

To understand the above formulas, let's try to calculate the surface area of ​​a cylinder using examples.

1. The radius of the base of the cylinder is 2, the height is 3. Determine the area of ​​the side surface of the cylinder.

The total surface area is calculated by the formula: S side. = 2prh

S side = 2 * 3.14 * 2 * 34.6 . Total ratings received: 990.