How to calculate the cross-sectional area of ​​a cylindrical part. Cylinder, cylinder area

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. The measurement of arable land within the new boundaries after the spill was carried out by special people. It was as a result of their activities that the new science, developed in Ancient Greece. There she received the name, and acquired practically 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.

Examples of the presence of cylindrical objects in Everyday life enough. 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.

Aggregate an infinite number generatrix - nothing more than the lateral surface of the cylinder - one of the elements of this 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 area of ​​​​the base (there are two of them) and making simple algebraic transformations, we obtain the final expression for determining the surface area of ​​\u200b\u200bthe 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.

A cylinder is a symmetrical spatial figure, the properties of which are considered in the senior grades of the school in the course of solid geometry. To describe it, such linear characteristics as the height and radius of the base are used. In this article, we will consider questions regarding what is the axial section of a cylinder, and how to calculate its parameters through the main 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 a parallel displacement of a segment of a fixed length along a certain curve. The main condition for this movement is that the segment of the plane of the curve should not belong.

The figure below shows a cylinder whose curve (guide) is an ellipse.

Here a segment of length h is its generatrix and its 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 the points of the generating lines.

Before proceeding to the consideration of the axial section of the cylinders, we will tell you what types of these figures are.

If the generating line is perpendicular to the bases of the figure, then they speak of a straight cylinder. Otherwise, the cylinder will be inclined. If you connect the central points of the two bases, then the resulting straight line is called the axis of the figure. The following figure 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.

Axial section of a straight cylinder

An axial section is any section of a 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 a half of the cylinder, which was obtained as a 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 right circular cylinder is a rectangle. Its sides are the diameter d of the base and the height h of the figure.

We write formulas for the area of ​​the axial section of the cylinder and the length h d of its diagonal:

A rectangle has two diagonals, but both of them 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 an inclined cylinder made of paper. If you perform its axial section, then you will no longer get a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of a section of a straight cylinder, is equal to the diameter d of the base, while the other is the length of the generating segment. Let's denote it b.

To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. We also need an angle between them. Assume that the acute angle between the guide and the base is α. It will also be the angle between the sides of the parallelogram. Then the formula for the area of ​​the axial section of the 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 give without derivation expressions that allow us to calculate the diagonals of a parallelogram from known sides and sharp corner 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 how to use the acquired knowledge to solve the following problem. Let a round straight cylinder be given. It is known that the axial section of a cylinder is a square. What is the area of ​​this section if the entire figure is 100 cm 2?

To calculate the desired area, you must 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. Given this, 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 the square section is equal to the diameter of the base of the figure, then to calculate its area S, it will be valid following formula:

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.

Cylinder (derived from Greek, from the words "skating rink", "roller") is a geometric body, which is limited on the outside by a surface called a cylindrical one 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 obtained by a straight line in space. These movements are such that the selected point of this straight line moves along a flat-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. Cylinders are of several types:

1. Circular, straight cylinder. For such a cylinder, the base and the guide are perpendicular to the generatrix, and there is

2. Inclined cylinder. He has an angle between the generating line and the base is not straight.

3. A cylinder of a different shape. Hyperbolic, elliptical, parabolic and others.

The area of ​​the cylinder, as well as the area full surface of any cylinder is found by adding the areas of the bases of this figure and the area of ​​\u200b\u200bthe lateral surface.

The formula for calculating the total area of ​​a cylinder for a circular, straight cylinder is:

Sp = 2p Rh + 2p R2 = 2p R (h+R).

The area of ​​the lateral surface is a little more difficult to find than the area of ​​the entire cylinder; it is calculated by multiplying the length of the generatrix by the perimeter of the section formed by the plane that is perpendicular to the generatrix.

The cylinder data for a circular, straight cylinder is recognized by the development of this object.

A development is a rectangle that has height h and length P, which is equal to the perimeter of the base.

Hence it follows that side area cylinder is equal to the area of ​​the sweep 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 lateral surface area should be equal to the product of the length of its generatrix and the perimeter of the section, which is perpendicular to this generatrix.

Unfortunately, there is no simple formula for expressing the lateral surface area of ​​an inclined cylinder in terms of its height and its base parameters.

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, respectively, 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 this figure by 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 ​​\u200b\u200bthis section will be equal to the product of the height of this cylinder and a certain chord. To get a chord, you need to build a circle at the base of the cylinder, draw a radius and set aside 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 searched for 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 ​​\u200b\u200bthis section is found as the area of ​​​​the circle.

The area of ​​a circle is:

S env. = 2p R2.

To find R, you need to divide its length C by 2p:

R = C \ 2n, where n is pi, a mathematical constant calculated to work with circle data and equal to 3.14.

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 it side surface around 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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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: