Tutorial: Cylinder. The simplest sections of a cylinder Called a cylinder axial section square
A cylinder is a symmetrical spatial figure, the properties of which are considered in high school in the course of stereometry. To describe it, linear characteristics such as height and base radius are used. In this article we will consider questions regarding what the axial section of a cylinder is and how to calculate its parameters through the basic linear characteristics of the figure.
Geometric figure
First, let's define the figure that will be discussed in the article. A cylinder is a surface formed by parallel movement of a segment of a fixed length along a certain curve. The main condition for this movement is that the segment should not belong to the plane of the curve.
The figure below shows a cylinder whose curve (guide) is an ellipse.
Here a segment of length h is its generator and height.
It can be seen that the cylinder consists of two identical bases (ellipses in this case), which lie in parallel planes, and a side surface. The latter belongs to all points of the forming lines.
Before moving on to considering the axial section of cylinders, we will tell you what types of these figures there are.
If the generating line is perpendicular to the bases of the figure, then we speak of a straight cylinder. Otherwise the cylinder will be inclined. If you connect the central points of two bases, the resulting straight line is called the axis of the figure. The figure below shows the difference between straight and inclined cylinders.
It can be seen that for a straight figure, the length of the generating segment coincides with the value of the height h. For an inclined cylinder, the height, that is, the distance between the bases, is always less than the length of the generatrix line.
Axial section of a straight cylinder
Axial is any section of the cylinder that contains its axis. This definition means that the axial section will always be parallel to the generatrix.
In a straight cylinder, the axis passes through the center of the circle and is perpendicular to its plane. This means that the circle under consideration will intersect along its diameter. The figure shows half a cylinder, which is the result of the intersection of the figure with a plane passing through the axis.
It is not difficult to understand that the axial section of a straight circular cylinder is a rectangle. Its sides are the diameter d of the base and the height h of the figure.
Let us write the formulas for the axial cross-sectional area of the cylinder and the length h d of its diagonal:
A rectangle has two diagonals, but both are equal to each other. If the radius of the base is known, then it is not difficult to rewrite these formulas through it, given that it is half the diameter.
Axial section of an inclined cylinder
The picture above shows a slanted cylinder made of paper. If you make its axial section, you will no longer get a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of the cross-section of a straight cylinder, is equal to the diameter d of the base, the other is the length of the forming segment. Let's denote it b.
To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. Another angle between them is needed. Let us assume that the acute angle between the guide and the base is α. This will also be the angle between the sides of the parallelogram. Then the formula for the axial cross-sectional area of an inclined cylinder can be written as follows:
The diagonals of the axial section of an inclined cylinder are somewhat more difficult to calculate. A parallelogram has two diagonals of different lengths. We present expressions without derivation that allow us to calculate the diagonals of a parallelogram using known sides and the acute angle between them:
l 1 = √(d 2 + b 2 - 2*b*d*cos(α));
l 2 = √(d 2 + b 2 + 2*b*d*cos(α))
Here l 1 and l 2 are the lengths of the small and large diagonals, respectively. These formulas can be obtained independently if we consider each diagonal as a vector by introducing a rectangular coordinate system on the plane.
Straight Cylinder Problem
We will show you how to use the knowledge gained to solve the following problem. Let us be given a round straight cylinder. It is known that the axial cross section of a cylinder is square. What is the area of this section if the entire figure is 100 cm 2?
To calculate the required area, you need to find either the radius or the diameter of the base of the cylinder. To do this, we use the formula for the total area S f of the figure:
Since the axial section is a square, this means that the radius r of the base is half the height h. Taking this into account, we can rewrite the equality above as:
S f = 2*pi*r*(r + 2*r) = 6*pi*r 2
Now we can express the radius r, we have:
Since the side of a square section is equal to the diameter of the base of the figure, the following formula will be valid to calculate its area S:
S = (2*r) 2 = 4*r 2 = 2*S f / (3*pi)
We see that the required area is uniquely determined by the surface area of the cylinder. Substituting the data into equality, we come to the answer: S = 21.23 cm 2.
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.
1. Axial section cylinder is a section of the cylinder by a plane passing through its axis. The axial cross section of the cylinder is rectangle.
2. Section of a cylinder with a plane parallel to the base.
In this case, the cross-section is a circle equal and parallel to the base.
Cone
A cone is a geometric body that consists of a circle - grounds cone, a point not lying in the plane of this circle, − peaks cone and all segments connecting the top of the cone with the points of the base.
The segments connecting the vertex of the cone with the points of the base circle are called forming cone
The cone is called direct, if the straight line connecting the top of the cone with the center of the base is perpendicular to the plane of the base.
On rice. A) straight cone, b) inclined cone.
In what follows, we will only consider a straight cone!
S- the top of the cone.
Circle with centers ABOUT– the base of the cone.
S.A.,C.B., SC– forming cones.
Height of a cone is called the perpendicular descended from its apex to the plane of the base.
Axis of a cone is called a straight line containing its height ( SO).
Cone properties:
The generators of the cone are equal.
A cone can be considered as a body obtained by rotating a right triangle around its side.
The simplest sections of a cone.
1. Axial section cone is a section of a cone by a plane passing through its axis. The axial section of the cone is triangle.
2. Section of a cone with a plane parallel to the base.
In this case, the cross-section is a circle similar to and parallel to the base.
A ball is a geometric body that consists of all points in space located at a distance not greater than a given one from a given point.
This point ( ABOUT) is called center ball, and this distance is radius ball.
The boundary of the ball is called spherical surface or sphere.
Any segment connecting the center of a ball to a point on the spherical surface is called radius ball ( O.D., OB, OA).
Ball diameter is a segment connecting two points on a spherical surface and passing through the center of the ball ( AB).
Ball properties:
The radii of the ball are equal;
The diameters of the ball are equal.
A ball can be considered as a body obtained by rotating a semicircle around its diameter.
The simplest sections of a ball
1. Section of a ball by a plane passing through its center. In this case, the section is big circle.
2. Section of a ball by a plane Not passing through its center. In this case, the section is circle.
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