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 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 the two bases of the cylinder to the area of the lateral surface of the cylinder, we get the total surface area of the cylinder
S full \u003d 2πRH + 2πR 2 \u003d 2πR (H + R).
Straight cylinder volume
Theorem. The volume of a right cylinder is equal to the product of the area of its base and the height , i.e.where Q is the base area and H is the height of the cylinder.
Since the area of the base of the cylinder is Q, 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 sequences of prisms whose bases are the described and inscribed polygons considered above, and whose lateral edges are parallel to the generatrix of the given cylinder and have length H. These prisms are described 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 \u003d π R 2, and therefore
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 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 a regular tin, which does not have top cover 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
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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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. After the spill, special people were engaged in measuring the areas of arable land within the new boundaries. It was as a result of their activities that a new science arose, which was 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.
There are plenty of examples of the presence of cylindrical objects in everyday life. 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.
The totality of an infinite set of generators is nothing more than the lateral surface of a cylinder - one of the elements of a given 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 rotation of a 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.
When studying stereometry, one of the main topics is the "Cylinder". The lateral surface area is considered, if not the main, then an important formula in solving geometric problems. However, it is important to remember the definitions that will help you navigate the examples and when proving various theorems.
The concept of a cylinder
First, we need to consider a few definitions. Only after studying them can one begin to consider the question of the formula for the area of the lateral surface of a cylinder. Based on this entry, other expressions can be calculated.
- A cylindrical surface is a plane described by a generatrix that moves and remains parallel. given direction sliding along the existing curve.
- There is also a second definition: a cylindrical surface is formed by a set of parallel lines intersecting a given curve.
- The generatrix is conventionally called the height of the cylinder. When it moves around an axis passing through the center of the base, a designated geometric body is obtained.
- An axis is a straight line passing through both bases of the figure.
- A cylinder is a stereometric body bounded by an intersecting lateral surface and 2 parallel planes.
There are varieties of this three-dimensional figure:
- By circular is meant a cylinder, the guide of which is a circle. Its main components are the radius of the base and the generatrix. The latter is equal to the height of the figure.
- There is a straight cylinder. It got its name due to the perpendicularity of the generatrix to the bases of the figure.
- The third type is a beveled cylinder. In textbooks, you can also find another name for it - "circular cylinder with a beveled base." This figure is determined by the radius of the base, the minimum and maximum height.
- An equilateral cylinder is understood as a body having equal height and diameter of a circular plane.
Conventions
Traditionally, the main "components" of a cylinder are called as follows:
- The radius of the base is R (it also replaces the similar value of the stereometric figure).
- Generating - L.
- Height - H.
- The base area is S main (in other words, you need to find the specified circle parameter).
- Beveled cylinder heights - h 1, h 2 (minimum and maximum).
- The lateral surface area is S side (if you unfold it, you get a kind of rectangle).
- The volume of a stereometric figure is V.
- Total surface area - S.
"Components" of a stereometric figure
When studying a cylinder, the lateral surface area plays an important role. This is related to the fact that given formula included in several others, more complex. Therefore, it is necessary to be well versed in theory.
The main components of the figure are:
- Side surface. As you know, it is obtained due to the movement of the generatrix along a given curve.
- The complete surface includes the existing bases and the side plane.
- The section of the cylinder, as a rule, is a rectangle located parallel to the axis of the figure. Otherwise, it is called a plane. It turns out that the length and width are part-time components of other figures. So, conditionally, the lengths of the section are generators. Width - parallel chords of a stereometric figure.
- By axial section is meant the location of the plane through the center of the body.
- And finally, the final definition. A tangent is a plane passing through the generatrix of the cylinder and at right angles to the axial section. In this case, one condition must be met. The specified generatrix must be included in the plane of the axial section.
Basic formulas for working with a cylinder
In order to answer the question of how to find the surface area of a cylinder, it is necessary to study the main "components" of a stereometric figure and the formulas for finding them.
These formulas differ in that first the expressions for the beveled cylinder are given, and then for the straight one.
Broken Solution Examples
You need to find the area of the lateral surface of the cylinder. The diagonal of the section AC = 8 cm is given (moreover, it is axial). When in contact with the generatrix, it turns out< ACD = 30°
Solution. Since the values of the diagonal and the angle are known, then in this case:
- CD = AC*cos 30°.
A comment. Triangle ACD, in specific example, rectangular. This means that the quotient of dividing CD and AC = the cosine of the given angle. The value of trigonometric functions can be found in a special table.
Similarly, you can find the AD value:
- AD = AC*sin 30°
Now you need to calculate the desired result using the following formulation: the area of \u200b\u200bthe lateral surface of the cylinder is equal to twice the result of multiplying "pi", the radius of the figure and its height. Another formula should also be used: the area of the base of the cylinder. It is equal to the result of multiplying "pi" by the square of the radius. And finally, the last formula: total surface area. It is equal to the sum of the previous two areas.
given cylinders. Their volume = 128 * n cm³. Which cylinder has the smallest total area?
Solution. First you need to use the formulas for finding the volume of a figure and its height.
Since the total surface area of the cylinder is known from theory, it is necessary to apply its formula.
If we consider the resulting formula as a function of the area of the cylinder, then the minimum “exponent” will be reached at the extremum point. To get the last value, you need to use differentiation.
Formulas can be viewed in a special table for finding derivatives. In the future, the result found is equated to zero and the solution of the equation is found.
Answer: S min will be reached at h = 1/32 cm, R = 64 cm.
A stereometric figure is given - a cylinder and a section. The latter is carried out in such a way that it is located parallel to the axis of the stereometric body. The cylinder has the following parameters: VK = 17 cm, h = 15 cm, R = 5 cm. It is necessary to find the distance between the section and the axis.
Since the cross section of a cylinder is understood to be VSKM, i.e., a rectangle, then its side ВМ = h. WMC needs to be considered. The triangle is rectangular. Based on this statement, we can deduce the correct assumption that MK = BC.
VK² = VM² + MK²
MK² = VK² - VM²
MK² = 17² - 15²
From this we can conclude that MK \u003d BC \u003d 8 cm.
The next step is to draw a section through the base of the figure. It is necessary to consider the resulting plane.
AD is the diameter of the stereometric figure. It is parallel to the section mentioned in the problem statement.
BC is a straight line located on the plane of the existing rectangle.
ABCD is a trapezoid. In a particular case, it is considered isosceles, since a circle is described around it.
If you find the height of the resulting trapezoid, then you can get the answer given at the beginning of the problem. Namely: finding the distance between the axis and the drawn section.
To do this, you need to find the values of AD and OS.
Answer: the section is located 3 cm from the axis.
Tasks for fixing the material
Given a cylinder. The lateral surface area is used in the further solution. Other options are known. The area of the base is Q, the area of the axial section is M. It is necessary to find S. In other words, the total area of the cylinder.
Given a cylinder. The lateral surface area must be found in one of the steps of solving the problem. It is known that height = 4 cm, radius = 2 cm. It is necessary to find the total area of the stereometric figure.
