A cylinder (comes from the Greek language, from the words “roller”, “roller”) is a geometric body that is limited on the outside by a surface called cylindrical and two planes. These planes intersect the surface of the figure and are parallel to each other.
A cylindrical surface is a surface that is formed by a straight line in space. These movements are such that the selected point of this straight line moves along a plane type curve. Such a straight line is called a generatrix, and a curved line is called a guide.
The cylinder consists of a pair of bases and a lateral cylindrical surface. There are several types of cylinders:
1. Circular, straight cylinder. Such a cylinder has a base and guide perpendicular to the generating line, and there is
2. Inclined cylinder. Its angle between the generating line and the base is not straight.
3. A cylinder of a different shape. Hyperbolic, elliptic, parabolic and others.
The area of a cylinder, as well as the total surface area of any cylinder, is found by adding the areas of the bases of this figure and the area of the lateral surface.
The formula for calculating the total area of the cylinder for a circular, straight cylinder:
Sp = 2p Rh + 2p R2 = 2p R (h+R).
The area of the lateral surface is found to be a little more complicated than the area of the entire cylinder; it is calculated by multiplying the length of the generatrix line by the perimeter of the section formed by a plane that is perpendicular to the generatrix line.
The given cylinder for a circular, straight cylinder is recognized by the development of this object.
A development is a rectangle that has a height h and a length P, which is equal to the perimeter of the base.
It follows that the lateral area of the cylinder is equal to the sweep area and can be calculated using this formula:
If we take a circular, straight cylinder, then for it:
P = 2p R, and Sb = 2p Rh.
If the cylinder is inclined, then the area of the lateral surface should be equal to the product of the length of its generating line and the perimeter of the section, which is perpendicular to this generating line.
Unfortunately, there is no simple formula for expressing the lateral surface area of an inclined cylinder in terms of its height and the parameters of its base.
To calculate a cylinder, you need to know a few facts. If a section with its plane intersects the bases, then such a section is always a rectangle. But these rectangles will be different, depending on the position of the section. One of the sides of the axial section of the figure, which is perpendicular to the bases, is equal to the height, and the other is equal to the diameter of the base of the cylinder. And the area of such a section, accordingly, is equal to the product of one side of the rectangle by the other, perpendicular to the first, or the product of the height of a given figure and the diameter of its base.
If the section is perpendicular to the bases of the figure, but does not pass through the axis of rotation, then the area of this section will be equal to the product of the height of this cylinder and a certain chord. To get a chord, you need to construct a circle at the base of the cylinder, draw a radius and plot on it the distance at which the section is located. And from this point you need to draw perpendiculars to the radius from the intersection with the circle. The intersection points are connected to the center. And the base of the triangle is the desired one, which is sought by sounds like this: “The sum of the squares of two legs is equal to the hypotenuse squared”:
C2 = A2 + B2.
If the section does not affect the base of the cylinder, and the cylinder itself is circular and straight, then the area of this section is found as the area of the circle.
The area of the circle is:
S env. = 2п R2.
To find R, you need to divide its length C by 2n:
R = C\2n, where n is pi, a mathematical constant calculated to work with circle data and equal to 3.14.
Stereometry is a branch of geometry in which figures in space are studied. The main figures in space are a point, a straight line and a plane. In stereometry, a new type of relative arrangement of lines appears: crossing lines. This is one of the few significant differences between stereometry and planimetry, since in many cases problems in stereometry are solved by considering various planes in which planimetric laws are satisfied.
In the nature around us, there are many objects that are physical models of this figure. For example, many machine parts have the shape of a cylinder or are some combination thereof, and the majestic columns of temples and cathedrals, made in the shape of cylinders, emphasize their harmony and beauty.
Greek − kylindros. An ancient term. In everyday life - a papyrus scroll, a roller, a roller (verb - to twist, roll).
For Euclid, a cylinder is obtained by rotating a rectangle. In Cavalieri - by the movement of the generatrix (with an arbitrary guide - a “cylinder”).
The purpose of this essay is to consider a geometric body - a cylinder.
To achieve this goal, it is necessary to consider the following tasks:
− give definitions of a cylinder;
− consider the elements of the cylinder;
− study the properties of the cylinder;
− consider the types of cylinder sections;
− derive the formula for the area of a cylinder;
− derive the formula for the volume of a cylinder;
− solve problems using a cylinder.
1.1. Definition of a cylinder
Let us consider some line (curve, broken or mixed) l lying in some plane α, and some straight line S intersecting this plane. Through all points of a given line l we draw straight lines parallel to straight line S; the surface α formed by these straight lines is called a cylindrical surface. Line l is called the guide of this surface, lines s 1, s 2, s 3,... are its generators.
If the guide is broken, then such a cylindrical surface consists of a number of flat strips enclosed between pairs of parallel straight lines, and is called a prismatic surface. The generatrices passing through the vertices of the guide broken line are called the edges of the prismatic surface, the flat strips between them are its faces.
If we cut any cylindrical surface with an arbitrary plane that is not parallel to its generators, we will obtain a line that can also be taken as a guide for this surface. Among the guides, the one that stands out is the one that is obtained by cutting the surface with a plane perpendicular to the generatrices of the surface. Such a section is called a normal section, and the corresponding guide is called a normal guide.
If the guide is a closed (convex) line (broken or curved), then the corresponding surface is called a closed (convex) prismatic or cylindrical surface. The simplest of cylindrical surfaces has a circle as its normal guide. Let us dissect a closed convex prismatic surface with two planes parallel to each other, but not parallel to the generators.
In sections we obtain convex polygons. Now the part of the prismatic surface enclosed between the planes α and α" and the two polygonal plates formed in these planes limit a body called a prismatic body - a prism.
Cylindrical body - a cylinder is defined similarly to a prism:
A cylinder is a body bounded on the sides by a closed (convex) cylindrical surface, and on the ends by two flat parallel bases. Both bases of the cylinder are equal, and all the constituents of the cylinder are also equal, i.e. segments of the generatrices of a cylindrical surface between the planes of the bases.
A cylinder (more precisely, a circular cylinder) is a geometric body that consists of two circles that do not lie in the same plane and are combined by parallel translation, and all the segments connecting the corresponding points of these circles (Fig. 1).
The circles are called the bases of the cylinder, and the segments connecting the corresponding points of the circles' circumferences are called the generators of the cylinder.
Since parallel translation is motion, the bases of the cylinder are equal.
Since during parallel translation the plane transforms into a parallel plane (or into itself), then the bases of the cylinder lie in parallel planes.
Since during parallel translation the points are shifted along parallel (or coinciding) lines by the same distance, then the generators of the cylinder are parallel and equal.
The surface of the cylinder consists of the base and side surface. The lateral surface is composed of generatrices.
A cylinder is called straight if its generators are perpendicular to the planes of the bases.
A straight cylinder can be visually imagined as a geometric body that describes a rectangle when rotating it around its side as an axis (Fig. 2).
Rice. 2 − Straight cylinder
In what follows, we will consider only the straight cylinder, calling it simply a cylinder for brevity.
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. It is parallel to the generators.
A cylinder is called equilateral if its height is equal to the diameter of the base.
If the bases of the cylinder are flat (and, therefore, the planes containing them are parallel), then the cylinder is said to stand on a 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 we speak of a circular (circular) cylinder; if it’s an ellipse, then it’s elliptical.
1. 3. Sections of the cylinder
The cross section of a cylinder with a plane parallel to its axis is a rectangle (Fig. 3, a). Its two sides are the generators of the cylinder, and the other two are parallel chords of the bases.
A) 
b)
V) G)
Rice. 3 – Sections of the cylinder
In particular, the rectangle is the axial section. This is a section of a cylinder with a plane passing through its axis (Fig. 3, b).
The cross section of a cylinder with a plane parallel to the base is a circle (Figure 3, c).
The cross section of a cylinder with a plane not parallel to the base and its axis is an oval (Fig. 3d).
Theorem 1. A plane parallel to the plane of the base of the cylinder intersects its lateral surface along a circle equal to the circumference of the base.
Proof. Let β be a plane parallel to the plane of the base of the cylinder. Parallel translation in the direction of the cylinder axis, combining plane β with the plane of the base of the cylinder, combines the section of the side surface by plane β with the circumference of the base. The theorem has been proven.
The lateral surface area of the cylinder.
The area of the lateral surface of the cylinder is taken to be the limit to which the area of the lateral surface of a regular prism inscribed in the cylinder tends when the number of sides of the base of this prism increases indefinitely.
Theorem 2. The area of the lateral surface of a cylinder is equal to the product of the circumference of its base and its height (S side.c = 2πRH, where R is the radius of the base of the cylinder, H is the height of the cylinder).
A) 
b) 
Rice. 4 − Cylinder lateral surface area
Proof.
Let P n and H be the perimeter of the base and the height of a regular n-gonal prism inscribed in the cylinder, respectively (Fig. 4, a). Then the area of the lateral surface of this prism is S side.c − P n H. Let us assume that the number of sides of the polygon inscribed in the base grows without limit (Fig. 4, b). Then the perimeter P n tends to the circumference C = 2πR, where R is the radius of the base of the cylinder, and the height H does not change. Thus, the area of the lateral surface of the prism tends to the limit of 2πRH, i.e., the area of the lateral surface of the cylinder is equal to S side.c = 2πRH. The theorem has been proven.
The total surface area of the cylinder.
The total surface area of a cylinder is the sum of the areas of the lateral surface and the two bases. The area of each base of the cylinder is equal to πR 2, therefore, the area of the total surface of the cylinder S total is calculated by the formula S side.c = 2πRH+ 2πR 2.
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Rice. 5 − Total surface area of the cylinder
If the side surface of the cylinder is cut along the generatrix FT (Fig. 5, a) and unfolded so that all the generators are in the same plane, then as a result we get a rectangle FTT1F1, which is called the development of the side surface of the cylinder. Side FF1 of the rectangle is the development of the circle of the base of the cylinder, therefore, FF1=2πR, and its side FT is equal to the generatrix of the cylinder, i.e. FT = H (Fig. 5, b). Thus, the area FT∙FF1=2πRH of the cylinder development is equal to the area of its lateral surface.
1.5. Cylinder volume
If a geometric body is simple, that is, it can be divided into a finite number of triangular pyramids, then its volume is equal to the sum of the volumes of these pyramids. For an arbitrary body, the volume is determined as follows.
A given body has a volume V if there are simple bodies containing it and simple bodies contained in it with volumes as little different from V as desired.
Let us apply this definition to finding the volume of a cylinder with base radius R and height H.
When deriving the formula for the area of a circle, two n-gons were constructed (one containing the circle, the other contained in the circle) such that their areas, with an unlimited increase in n, approached the area of the circle without limit. Let's construct such polygons for the circle at the base of the cylinder. Let P be a polygon containing a circle, and P" be a polygon contained in a circle (Fig. 6).
Rice. 7 − Cylinder with a prism described and inscribed in it
Let us construct two straight prisms with bases P and P" and a height H equal to the height of the cylinder. The first prism contains a cylinder, and the second prism is contained in a cylinder. Since with an unlimited increase in n, the areas of the bases of the prisms unlimitedly approach the area of the base of the cylinder S, then their volumes approach SH without limit. According to the definition, the volume of a cylinder
V = SH = πR 2 H.
So, the volume of a cylinder is equal to the product of the area of the base and the height.
Task 1.
The axial section of the cylinder is a square with area Q.
Find the area of the base of the cylinder.
Given: cylinder, square - axial section of the cylinder, S square = Q.
Find: S main cylinder
The side of the square is . It is equal to the diameter of the base. Therefore the area of the base is 
.
Answer: S main cylinder. =
Task 2.
A regular hexagonal prism is inscribed in a cylinder. Find the angle between the diagonal of its side face and the axis of the cylinder if the radius of the base is equal to the height of the cylinder.
Given: cylinder, regular hexagonal prism inscribed in the cylinder, base radius = height of the cylinder.
Find: the angle between the diagonal of its side face and the axis of the cylinder.
Solution: The lateral faces of the prism are squares, since the side of a regular hexagon inscribed in a circle is equal to the radius.
The edges of the prism are parallel to the cylinder axis, therefore the angle between the diagonal of the face and the cylinder axis is equal to the angle between the diagonal and the side edge. And this angle is 45°, since the faces are squares.
Answer: the angle between the diagonal of its side face and the axis of the cylinder = 45°.
Task 3.
The height of the cylinder is 6 cm, the radius of the base is 5 cm.
Find the area of a section drawn parallel to the cylinder axis at a distance of 4 cm from it.
Given: H = 6cm, R = 5cm, OE = 4cm.
Find: S sec.
S sec. = KM×KS,
OE = 4 cm, KS = 6 cm.
Triangle OKM - isosceles (OK = OM = R = 5 cm),
triangle OEK is a right triangle.
From the triangle OEK, according to the Pythagorean theorem:
KM = 2EK = 2×3 = 6,
S sec. = 6×6 = 36 cm 2.
The purpose of this essay has been fulfilled; a geometric body such as a cylinder has been considered.
The following tasks are considered:
− the definition of a cylinder is given;
− the elements of the cylinder are considered;
− the properties of the cylinder were studied;
− types of cylinder sections are considered;
− the formula for the area of a cylinder is derived;
− the formula for the volume of a cylinder is derived;
− solved problems using a cylinder.
1. Pogorelov A.V. Geometry: Textbook for 10 – 11 grades of educational institutions, 1995.
2. Beskin L.N. Stereometry. Manual for secondary school teachers, 1999.
3. Atanasyan L. S., Butuzov V. F., Kadomtsev S. B., Kiseleva L. S., Poznyak E. G. Geometry: Textbook for grades 10 - 11 of educational institutions, 2000.
4. Aleksandrov A.D., Werner A.L., Ryzhik V.I. Geometry: textbook for grades 10-11 in general education institutions, 1998.
5. Kiselev A. P., Rybkin N. A. Geometry: Stereometry: grades 10 – 11: Textbook and problem book, 2000.
The area of each base of the cylinder is π r 2, the area of both bases will be 2π r 2 (fig.).The area of the lateral surface of a cylinder is equal to the area of a rectangle whose base is 2π r, and the height is equal to the height of the cylinder h, i.e. 2π rh.
The total surface of the cylinder will be: 2π r 2 + 2π rh= 2π r(r+ h).
The area of the lateral surface of the cylinder is taken to be sweep area its lateral surface.
Therefore, the area of the lateral surface of a right circular cylinder is equal to the area of the corresponding rectangle (Fig.) and is calculated by the formula
S b.c. = 2πRH, (1)
If we add the area of its two bases to the area of the lateral surface of the cylinder, we obtain the total surface area of the cylinder
S full =2πRH + 2πR 2 = 2πR (H + R).
Volume of a straight cylinder
Theorem. The volume of a straight cylinder is equal to the product of the area of its base and its height , i.e.where Q is the area of the base, and H is the height of the cylinder.
Since the area of the base of the cylinder is Q, then there are sequences of circumscribed and inscribed polygons with areas Q n and Q' n such that
\(\lim_(n \rightarrow \infty)\) Q n= \(\lim_(n \rightarrow \infty)\) Q’ n= Q.
Let us construct a sequence of prisms whose bases are the described and inscribed polygons discussed above, and whose side edges are parallel to the generatrix of the given cylinder and have length H. These prisms are circumscribed and inscribed for the given cylinder. Their volumes are found by the formulas
V n= Q n H and V' n= Q' n H.
Hence,
V= \(\lim_(n \rightarrow \infty)\) Q n H = \(\lim_(n \rightarrow \infty)\) Q’ n H = QH.
Consequence.
The volume of a right circular cylinder is calculated by the formula
V = π R 2 H
where R is the radius of the base and H is the height of the cylinder.
Since the base of a circular cylinder is a circle of radius R, then Q = π R 2, and therefore
Cylinder (circular cylinder) is a body that consists of two circles, 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' circumferences 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 the cylinder consists of the base and side surface. The lateral surface is made up of generatrices.
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 cylinders - elliptic, hyperbolic, parabolic. A prism is also considered as a type 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 cross section of a cylinder with a plane passing through the cylinder axis 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 side 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 ribs form the cylinder. A prism is said to be circumscribed about a cylinder if its bases are equal polygons circumscribed about the bases of the cylinder. The planes of its faces touch the side surface of the cylinder.
The lateral surface area of a 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 straight cylinder can be found by its development. The development of a 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 S b = 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 the 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 = π R 2 h = π (d 2 / 4)h,
where d is the diameter of the base.
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