EntranceWhat geometric figure lies at the base of the cylinder. Geometric bodies
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 as an example.
A cylinder has three surfaces: the top, the base, and side surface.
The top and base of a cylinder are circles and are easy to identify.
It is known that the area of a circle is equal to πr 2. Therefore, the formula for the area of two circles (the top and base of the cylinder) will be πr 2 + πr 2 = 2πr 2.
The third, side surface of the cylinder, is the curved wall of the cylinder. In order to better imagine this surface, let's try to transform it to get a recognizable shape. Imagine that the cylinder is an ordinary tin can that does not have a top lid or bottom. Let's make a vertical cut on the side wall from the top to the bottom of the can (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).
After the resulting jar is fully opened, 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's return for a moment to the original cylinder. The vertex of the original cylinder is a circle, and we know that the circumference is calculated by the formula: L = 2πr. It is marked in red in the figure.
When the side wall of the cylinder is fully opened, 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 received a formula for calculating the area of the lateral surface of the cylinder.
Formula for the lateral surface area of a cylinder
S side = 2πrh
Total surface area of a cylinder
Finally, if we add the area of all three surfaces, we get the formula for the total surface area of a cylinder. The surface area of a 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 identical to the formula 2πr (r + h).
Formula for the total surface area of a cylinder
S = 2πr 2 + 2πrh = 2πr(r + h)
r – radius of the cylinder, h – 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 lateral surface of the cylinder.
The total surface area is calculated using the formula: S side. = 2πrh
S side = 2 * 3.14 * 2 * 34.6. Total ratings received: 990.
kýlindros, roller, roller) - a geometric body limited by a cylindrical surface (called the lateral surface of the cylinder) and no more than two surfaces (the bases of the cylinder); Moreover, if there are two bases, then one is obtained from the other by parallel transfer along the generatrix of the side surface of the cylinder; and the base intersects each generatrix of the lateral surface exactly once.An infinite body bounded by a closed infinite cylindrical surface is called endless cylinder, bounded by a closed cylindrical beam and its base, is called open cylinder. The base and generators of a cylindrical beam are called the base and generators of an open cylinder, respectively.
A finite body bounded by a closed finite cylindrical surface and two sections that separate it is called end cylinder, or actually cylinder. The sections are called the bases of the cylinder. By the definition of a finite cylindrical surface, the bases of the cylinder are equal.
Obviously, the generatrices of the lateral surface of the cylinder are equal in length (called height cylinder) segments lying on parallel lines, and their ends lying on the bases of the cylinder. Mathematical curiosities include the definition of any finite three-dimensional surface without self-intersections as a cylinder of zero height (this surface is considered simultaneously as both bases of the finite cylinder). The bases of the cylinder affect the cylinder qualitatively.
If the bases of the cylinder are flat (and therefore the planes containing them are parallel), then the cylinder is called 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 we speak of a circular (circular) cylinder; if it’s an ellipse, then it’s elliptical.
The volume of the final cylinder is equal to the integral of the area of the base along the generatrix. In particular, the volume of a right circular cylinder is equal to
(where is the radius of the base, is the height).
The lateral surface area of the cylinder is calculated using the following formula:
The total surface area of a cylinder is the sum of the lateral surface area and the area of the bases. For a straight circular cylinder:
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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.
Plane, perpendicular to the axis cylinder, intersects its lateral surface along the circumference, equal circle grounds.
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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Let's start online new topic, and when I arrive we will conduct a test and test on the topic "Motion and Vectors".
- We are starting to get acquainted with a new class of geometric bodies - bodies of revolution. The first representative of this class that we meet is the cylinder.
- Why is a cylinder called a body of rotation?
C cylinder, obtained by rotating a rectangle around one of its sides.
- The cylinder consists of two circles and many segments.
- Cylinderis a geometric body consisting of two equal circles located in parallel planes and a set of segments connecting the corresponding points of these circles.
- Cylinder Element Definitions:
Cylinder bases– equal circles located in parallel planes
Cylinder height- This the distance between the planes of its bases.
Cylinder axis- this is a straight line passing through the centers of the base of the cylinder (the axis of the cylinder is the axis of rotation of the cylinder).
Axial section of the cylinder– section of the cylinder by a plane passing through the axis of the cylinder (the axial section of the cylinder is the plane of symmetry of the cylinder). All axial sections of the cylinder are equal rectangles
Generator of the cylinder- this is a segment connecting a point on the circle of the upper base with a corresponding point on the circle of the lower base. All generatrices are parallel to the axis of rotation and have the same length, equal to the height of the cylinder.
When rotating around an axis, the generatrix of the cylinder formslateral (cylindrical) surface of the cylinder.
Cylinder radiusis the radius of its base.
Straight cylinder- This is a cylinder, the generatrices of which are perpendicular to the base.
Equal cylinder– a cylinder whose height is equal to its diameter (show an equal cylinder: Use the button with the hand icon to switch the model back to interactive mode and change the height and radius values of the proposed model so that ).
- Derivation of the formula for the lateral surface area.
The development of the lateral surface of the cylinder is a rectangle with sidesH And C, Where His the height of the cylinder, andC– length of the base circumference. Let us obtain formulas for calculating the areas of the lateralS b and full S n surfaces: S b = H · C= 2π RH, S n = S b + 2 S= 2π R(R + H).
- R.S/ If anyone is interested, try following the link and have a look electronic resource about the cylinder, first on the page, install the OMS module on your PC and download the module. On the table that pops up, click play. And then look through all the pages in order.
- THANKS TO ALL.
Consolidation
Task No. 1. Calculate the lateral and total surface area of a cylinder whose radius is 3 cm and height 5 cm (pi and round the answer to whole numbers).
2. The height of the cylinder ish, base radiusR. Find the cross-sectional area of a plane drawn parallel to the cylinder axis at a distancea from her.
Homework: 522, 524, 526.
The name of the science “geometry” is translated as “earth measurement”. It originated through the efforts of the very first ancient land managers. And it happened like this: during the floods of the sacred Nile, streams of water sometimes washed away the boundaries of farmers’ plots, and the new boundaries might not coincide with the old ones. Taxes were paid by peasants to the treasury of the pharaoh in proportion to the size of the land allotment. Special people were involved in measuring the areas of arable land within the new boundaries after the spill. It was as a result of their activities that a new science arose, which was developed in Ancient Greece. There it received its name and practically acquired modern look. Subsequently, the term became an international name for the science of flat and volumetric figures Oh.
Planimetry is a branch of geometry dealing with the study flat figures. Another branch of science is stereometry, which examines the properties of spatial (volumetric) figures. Such figures include the one described in this article - a cylinder.
Examples of the presence of cylindrical objects in Everyday life plenty. Almost all rotating parts - shafts, bushings, journals, axles, etc. - have a cylindrical (much less often - conical) shape. The cylinder is also widely used in construction: towers, support columns, decorative columns. And also dishes, some types of packaging, pipes of various diameters. And finally - the famous hats, which have long become a symbol of male elegance. The list goes on and on.
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. These circles 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, something else, but not a cylinder) and are in parallel planes. The segments connecting corresponding points on circles are parallel and equal.
The set of an infinite number of forming elements is nothing more than the lateral surface of the 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 (in addition to circles) ellipses and other closed figures. But the cylinder may not necessarily have a closed shape. For example, the base of a cylinder can be a parabola, a hyperbola, or another open function. Such a cylinder will be open or deployed.
According to the angle of inclination of the cylinders forming the bases, they can be straight or inclined. For a straight cylinder, the generatrices are strictly perpendicular to the plane of the base. If this angle is different from 90°, the cylinder is inclined.
What is a surface of revolution
The straight circular cylinder is without a doubt the most common surface of rotation used in engineering. Sometimes, for technical reasons, conical, spherical, and some other types of surfaces are used, but 99% of all rotating shafts, axes, etc. are 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 certain straight line a, located vertically. ABCD is a rectangle, one of whose sides (segment AB) lies on a 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 rotating the figure - a rectangle - a cylinder is obtained. By rotating a triangle, you can get a cone, by rotating a semicircle - a ball, etc.
Cylinder surface area
In order to calculate the surface area of an ordinary right circular cylinder, it is necessary to calculate the areas of the bases and lateral surfaces.
First, let's look at how the lateral surface area is calculated. This is the product of the circumference of the cylinder and the height of the cylinder. The circumference, in turn, is equal to twice the product of the universal number P by the radius of the circle.
The area of a circle is known to be equal to the product P per square radius. So, by adding the formulas for the area of the lateral surface with the double expression for the area of the base (there are two of them) and performing simple algebraic transformations, we obtain the final expression for determining the surface area of the cylinder.
Determining the volume of a figure
The volume of a cylinder is determined according to the standard scheme: the surface area of the base is multiplied by the height.
Thus, the final formula looks like this: the desired value is defined as the product of the height of the body by the universal number P and by the square of the radius of the base.
The resulting formula, it must be said, is applicable to solving the most unexpected problems. In the same way as the volume of the cylinder, for example, the volume of electrical wiring is determined. This may be necessary to calculate the mass of the wires.
The only difference in the formula is that instead of the radius of one cylinder there is the diameter of the wiring strand divided in half and the number of strands in the wire appears in the expression N. Also, instead of height, the length of the wire is used. In this way, the volume of the “cylinder” is calculated not just by one, but by the number of wires in the braid.
Such calculations are often required in practice. After all, a significant part of water containers are 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 necessary to calculate what the volume of an inclined cylinder is.
The difference is that the surface area of the base is not multiplied by the length of the generatrix, as in the case of a straight cylinder, but by the distance between the planes - a perpendicular segment constructed between them.
As can be seen from the figure, such a segment is equal to the product of the length of the generatrix and the sine of the angle of inclination of the generatrix to the plane.
How to build a cylinder development
In some cases, it is necessary to cut out a cylinder ream. The figure below shows the rules by which a blank is constructed for the manufacture of a cylinder with a given height and diameter.
Please note that the drawing is shown without seams.
Differences between a beveled cylinder
Let us imagine a certain 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 not parallel to the first plane.
The figure shows a beveled cylinder. Plane A at a certain angle, different from 90° to the generators, intersects the figure.
This geometric shape is more often found 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 a beveled cylinder
The tilt of one of the planes of a beveled cylinder slightly changes the procedure for calculating both the surface area of such a figure and its volume.