Lateral area of ​​the cylinder formula. Tutorial: Cylinder

A cylinder is a figure consisting of a cylindrical surface and two circles arranged in parallel. Calculating the area of ​​a cylinder is a problem in the geometric branch of mathematics, which is solved quite simply. There are several methods for solving it, which as a result always come down to one formula.

How to find the area of ​​a cylinder - calculation rules

• To find out the area of ​​\u200b\u200bthe cylinder, you need to add two base areas with the area of ​​\u200b\u200bthe lateral surface: S \u003d S side. + 2 S main. In a more detailed version, this formula looks like this: S= 2 π rh+ 2 π r2= 2 π r(h+ r).
• The lateral surface area of ​​a given geometric body can be calculated if its height and the radius of the circle underlying the base are known. In this case, you can express the radius from the circumference, if it is given. The height can be found if the value of the generatrix is ​​specified in the condition. In this case, the generatrix will be equal to the height. The formula for the lateral surface of a given body looks like this: S= 2 π rh.
• The area of ​​the base is calculated by the formula for finding the area of ​​a circle: S osn= π r 2 . In some problems, the radius may not be given, but the circumference is given. With this formula, the radius is expressed quite easily. С=2π r, r= С/2π. It must also be remembered that the radius is half the diameter.
• When performing all these calculations, the number π usually does not translate into 3.14159 ... You just need to add it next to numerical value, which was obtained as a result of the calculations.
• Further, it is only necessary to multiply the found area of ​​\u200b\u200bthe base by 2 and add to the resulting number the calculated area of ​​\u200b\u200bthe lateral surface of the figure.
• If the problem indicates that the cylinder has an axial section and this is a rectangle, then the solution will be slightly different. In this case, the width of the rectangle will be the diameter of the circle that lies at the base of the body. The length of the figure will be equal to the generatrix or the height of the cylinder. It is necessary to calculate the desired values ​​\u200b\u200band substitute in an already known formula. In this case, the width of the rectangle must be divided by two to find the area of ​​the base. To find the side surface, the length is multiplied by two radii and by the number π.
• You can calculate the area of ​​a given geometric body through its volume. To do this, you need to derive the missing value from the formula V=π r 2 h.
• There is nothing difficult in calculating the area of ​​a cylinder. You only need to know the formulas and be able to derive from them the quantities necessary for the calculations.

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.

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:

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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Stereometry is a branch of geometry that studies shapes in space. The main figures in space are a point, a line and a plane. In stereometry appears the new kind mutual arrangement of lines: intersecting lines. This is one of the few significant differences between solid geometry and planimetry, since in many cases stereometry problems are solved by considering different 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 are in the form of a cylinder or some combination of them, and the majestic columns of temples and cathedrals, made in the form of cylinders, emphasize their harmony and beauty.

Greek − kyulindros. ancient term. In everyday life - a papyrus scroll, a roller, a skating rink (verb - twist, roll).

In Euclid, a cylinder is obtained by rotating a rectangle. For Cavalieri - by the movement of the generatrix (with an arbitrary guide - "cylinder").

The purpose of this essay is to consider a geometric body - a cylinder.

To achieve this goal, the following tasks should be considered:

− give definitions of a cylinder;

- consider the elements of the cylinder;

− to study the properties of the cylinder;

- consider the types of section of the cylinder;

- 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. Cylinder definition

Consider some line (curve, broken line or mixed line) l lying in some plane α and some straight line S intersecting this plane. Through all points of the given line l we draw lines parallel to the line S; the surface α formed by these straight lines is called a cylindrical surface. The line l is called the guide of this surface, the lines s 1 , s 2 , s 3 ,... are its generators.

If the guide is a broken line, then such a cylindrical surface consists of a series of flat strips enclosed between pairs of parallel lines, and is called a prismatic surface. The generatrices passing through the vertices of the guiding polyline are called the edges of the prismatic surface, the flat strips between them are called its faces.

If we cut any cylindrical surface with an arbitrary plane that is not parallel to its generators, then we get a line that can also be taken as a guide for this surface. Among the guides, one stands out, which is obtained from the section of the surface by a plane perpendicular to the generators 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 line or curve), then the corresponding surface is called a closed (convex) prismatic or cylindrical surface. Of the cylindrical surfaces, the simplest has its normal guide circle. Let us dissect a closed convex prismatic surface by two planes parallel to each other, but not parallel to the generators.

In the 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 the body, called the prismatic body - the prism.

A cylindrical body - a cylinder is defined similarly to a prism:
A cylinder is a body bounded laterally by a closed (convex) cylindrical surface, and from the ends by two flat parallel bases. Both bases of the cylinder are equal, and all generators of the cylinder are also equal to each other, i.e. segments forming a cylindrical surface between the planes of the bases.

A cylinder (more precisely, a circular cylinder) is a geometric body, which consists of two circles that do not lie in the same plane and are combined by parallel transfer, and all 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 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 translation the plane passes into a parallel plane (or into itself), then the bases of the cylinder lie in parallel planes.

Since, during parallel translation, the points are displaced along parallel (or coinciding) lines by the same distance, then 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 composed of generators.

A cylinder is called straight if its generators are perpendicular to the planes of the bases.

A straight cylinder can be visualized as a geometric body that describes a rectangle as it rotates around the side as an axis (Fig. 2).

Rice. 2 − Straight cylinder

In the following, we will consider only a 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 its base.

If the bases of the cylinder are flat (and hence the planes containing them are parallel), then the cylinder is said to be standing 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 one speaks of a circular (round) cylinder; if an ellipse, then elliptical.

1. 3. Sections of the cylinder

The section of the cylinder by a plane parallel to its axis is a rectangle (Fig. 3, a). Two of its sides are generatrices 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 the cylinder by a plane passing through its axis (Fig. 3, b).

The section of the cylinder by a plane parallel to the base is a circle (Fig. 3, c).

The cross section of the 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 axis of the cylinder, which combines the plane β with the plane of the base of the cylinder, combines the section of the side surface by the plane β with the circumference of the base. The theorem has been proven.

The area of ​​the lateral surface of the cylinder.

The area of ​​the side surface of the cylinder is taken as the limit to which the area of ​​the side surface tends right prism inscribed in a cylinder when the number of sides of the base of this prism increases indefinitely.

Theorem 2. The area of ​​the lateral surface of the cylinder is equal to the product of the circumference of its base and the 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 - The area of ​​the lateral surface of the cylinder

Proof.

Let P n and H, respectively, be the perimeter of the base and the height of a regular n-gonal prism inscribed in a cylinder (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 indefinitely (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 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 full surface of the cylinder S full is calculated by the formula S side.c \u003d 2πRH + 2πR 2.

Rice. 5 − Full 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 generatrix 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. The side FF1 of the rectangle is a development of the circumference 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 side surface.

1.5. Cylinder volume

If the geometric body is simple, that is, it can be divided into a finite number triangular pyramids, then its volume is equal to the sum of the volumes of these pyramids. For an arbitrary body, the volume is defined as follows.

A given body has volume V if there exist simple bodies and the simple bodies contained in it with volumes as little different from V as you like.

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 (one containing a circle, the other contained in a circle) were constructed such that their areas, with an unlimited increase in n, approached the area of ​​a circle indefinitely. Let us 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

We construct two straight prisms with bases P and P "and 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 indefinitely approach the area of ​​the base of the cylinder S, then their volumes indefinitely approach S H. 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 a cylinder is a square whose area is Q.

Find the area of ​​the base of the cylinder.

Given: cylinder, square - axial section of the cylinder, S square = Q.

Find: S main cyl.

The side of the square is . It is equal to the diameter of the base. So the area of ​​the base is .

Answer: S main cyl. =

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: a cylinder, a regular hexagonal prism inscribed in a cylinder, the radius of the base = the height of the cylinder.

Find: the angle between the diagonal of its side face and the axis of the cylinder.

Solution: Side faces prisms 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 axis of the cylinder, so the angle between the diagonal of the face and the axis of the cylinder 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 axis of the cylinder 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 OEK triangle, according to the Pythagorean theorem:

KM \u003d 2EK \u003d 2 × 3 \u003d 6,

S sec. \u003d 6 × 6 \u003d 36 cm 2.

The purpose of this essay is fulfilled, such a geometric body as a cylinder is considered.

The following tasks were considered:

− the definition of a cylinder is given;

− elements of the cylinder are considered;

− studied the properties of the cylinder;

− types of cylinder section are considered;

− the formula for the area of ​​a cylinder is derived;

− the formula for the volume of a cylinder is derived;

− Problems are solved with the use of a cylinder.

1. Pogorelov A. V. Geometry: A textbook for grades 10 - 11 of educational institutions, 1995.

2. Beskin L.N. Stereometry. Teacher's Guide high school, 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., Verner A.L., Ryzhik V.I. Geometry: textbook for grades 10-11 of educational institutions, 1998.

5. Kiselev A. P., Rybkin N. A. Geometry: Stereometry: Grades 10 - 11: Textbook and problem book, 2000.

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: top, bottom, and side surface.

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

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 * 3

S side = 6.28 * 6

S side = 37.68

The lateral surface area of ​​the cylinder is 37.68.

2. How to find the surface area of ​​a cylinder if the height is 4 and the radius is 6?

The total surface area is calculated by the formula: S = 2πr 2 + 2πrh

S = 2 * 3.14 * 6 2 + 2 * 3.14 * 6 * 4

S = 2 * 3.14 * 36 + 2 * 3.14 * 24

It is a geometric body bounded by two parallel planes and a cylindrical surface.

The cylinder consists of a side surface and two bases. The formula for the surface area of ​​a cylinder includes a separate calculation of the area of ​​the bases and the lateral surface. Since the bases in the cylinder are equal, then its total area will be calculated by the formula:

We will consider an example of calculating the area of ​​\u200b\u200ba cylinder after we know all the necessary formulas. First we need the formula for the area of ​​the base of a cylinder. Since the base of the cylinder is a circle, we need to apply:
We remember that these calculations use a constant number Π = 3.1415926, which is calculated as the ratio of the circumference of a circle to its diameter. This number is a mathematical constant. We will also consider an example of calculating the area of ​​​​the base of a cylinder a little later.

Cylinder side surface area

The formula for the area of ​​the lateral surface of a cylinder is the product of the length of the base and its height:

Now consider a problem in which we need to calculate the total area of ​​a cylinder. In a given figure, the height is h = 4 cm, r = 2 cm. Let's find the total area of ​​the cylinder.
First, let's calculate the area of ​​the bases:
Now consider an example of calculating the lateral surface area of ​​a cylinder. When expanded, it is a rectangle. Its area is calculated using the above formula. Substitute all the data into it:
The total area of ​​a circle is the sum of twice the area of ​​the base and the side:

Thus, using the formulas for the area of ​​the bases and the lateral surface of the figure, we were able to find the total surface area of ​​the cylinder.
The axial section of the cylinder is a rectangle in which the sides are equal to the height and diameter of the cylinder.

The formula for the area of ​​the axial section of a cylinder is derived from the calculation formula: