The area of ​​the cylinder is equal. Cylinder, cylinder area

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

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

• To find out the area of ​​the cylinder, you need to add the two areas of the base with the area of ​​the side surface: S = Sside + 2Sbase. 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 lying at its base are known. In this case, you can express the radius from the circumference, if given. The height can be found if the value of the generator is specified in the condition. In this case, the generatrix will be equal to the height. The formula for the lateral surface of this body looks like this: S= 2 π rh.
• The area of ​​the base is calculated using 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 may be given. With this formula, the radius is expressed quite easily. С=2π r, r= С/2π. You must also remember that the radius is half the diameter.
• When performing all these calculations, the number π is usually not translated into 3.14159... It just needs to be added next to the numerical value that was obtained as a result of the calculations.
• Next, you just need to multiply the found area of ​​the base by 2 and add to the resulting number the calculated area of ​​the lateral surface of the figure.
• If the problem indicates that the cylinder has an axial section and that it 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 lying at the base of the body. The length of the figure will be equal to the generatrix or height of the cylinder. It is necessary to calculate the required values ​​and substitute them into the 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 lateral surface, the length is multiplied by two radii and 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 complicated in calculating the area of ​​a cylinder. You just need to know the formulas and be able to derive from them the quantities necessary to carry out calculations.

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. The area of ​​arable land within the new boundaries was measured after the spill. special people. 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 three-dimensional figures.

Planimetry is a branch of geometry dealing with the study of plane 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.

There are plenty of examples of the presence of cylindrical objects in everyday life. 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 constituents is nothing more than lateral surface 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 (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 rotation - 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 ​​determining the lateral surface with the double 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 ​​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.

There are a large number of problems associated with the cylinder. In them you need to find the radius and height of the body or the type of its section. Plus, sometimes you need to calculate the area of ​​a cylinder and its volume.

Which body is a cylinder?

In the school curriculum, a circular cylinder, that is, a cylinder at the base, is studied. But the elliptical appearance of this figure is also distinguished. From the name it is clear that its base will be an ellipse or an oval.

The cylinder has two bases. They are equal to each other and are connected by segments that combine the corresponding points of the bases. They are called the generators of the cylinder. All generators are parallel to each other and equal. They make up the lateral surface of the body.

In general, a cylinder is an inclined body. If the generators make a right angle with the bases, then we speak of a straight figure.

Interestingly, a circular cylinder is a body of revolution. It is obtained by rotating a rectangle around one of its sides.

Main elements of the cylinder

The main elements of the cylinder look like this.

1. Height. It is the shortest distance between the bases of the cylinder. If it is straight, then the height coincides with the generatrix.
2. Radius. Coincides with the one that can be drawn at the base.
3. Axis. This is a straight line that contains the centers of both bases. The axis is always parallel to all generators. In a straight cylinder it is perpendicular to the bases.
4. Axial section. It is formed when a cylinder intersects a plane containing an axis.
5. Tangent plane. It passes through one of the generatrices and is perpendicular to the axial section, which is drawn through this generatrix.

How is a cylinder connected to a prism inscribed in it or described around it?

Sometimes there are problems in which you need to calculate the area of ​​a cylinder, but some elements of the associated prism are known. How do these figures relate?

If a prism is inscribed in a cylinder, then its bases are equal polygons. Moreover, they are inscribed in the corresponding bases of the cylinder. The lateral edges of the prism coincide with the generators.

The described prism has regular polygons at its base. They are described around the circles of the cylinder, which are its bases. The planes that contain the faces of the prism touch the cylinder along their generators.

On the area of ​​the lateral surface and base for a right circular cylinder

If you unwrap the side surface, you will get a rectangle. Its sides will coincide with the generatrix and the circumference of the base. Therefore, the lateral area of ​​the cylinder will be equal to the product of these two quantities. If you write down the formula, you get the following:

S side = l * n,

where n is the generator, l is the circumference.

Moreover, the last parameter is calculated using the formula:

l = 2 π * r,

here r is the radius of the circle, π is the number “pi” equal to 3.14.

Since the base is a circle, its area is calculated using the following expression:

S main = π * r 2 .

On the area of ​​the entire surface of a right circular cylinder

Since it is formed by two bases and a side surface, you need to add these three quantities. That is, the total area of ​​the cylinder will be calculated by the formula:

S floor = 2 π * r * n + 2 π * r 2 .

It is often written in a different form:

S floor = 2 π * r (n + r).

On the areas of an inclined circular cylinder

As for the bases, all the formulas are the same, because they are still circles. But the side surface no longer gives a rectangle.

To calculate the area of ​​the lateral surface of an inclined cylinder, you will need to multiply the values ​​of the generatrix and the perimeter of the section, which will be perpendicular to the selected generatrix.

The formula looks like this:

S side = x * P,

where x is the length of the cylinder generatrix, P is the perimeter of the section.

By the way, it is better to choose a section such that it forms an ellipse. Then the calculations of its perimeter will be simplified. The length of the ellipse is calculated using a formula that gives an approximate answer. But it is often sufficient for the tasks of a school course:

l = π * (a + b),

where “a” and “b” are the semi-axes of the ellipse, that is, the distance from the center to its nearest and farthest points.

The area of ​​the entire surface must be calculated using the following expression:

S floor = 2 π * r 2 + x * R.

What are some sections of a right circular cylinder?

When a section passes through an axis, its area is determined as the product of the generatrix and the diameter of the base. This is explained by the fact that it has the shape of a rectangle, the sides of which coincide with the designated elements.

To find the cross-sectional area of ​​a cylinder that is parallel to the axial one, you will also need a formula for a rectangle. In this situation, one of its sides will still coincide with the height, and the other will be equal to the chord of the base. The latter coincides with the section line along the base.

When the section is perpendicular to the axis, it looks like a circle. Moreover, its area is the same as that of the base of the figure.

It is also possible to intersect at some angle to the axis. Then the cross-section results in an oval or part of it.

Sample problems

Task No. 1. Given a straight cylinder whose base area is 12.56 cm 2 . It is necessary to calculate the total area of ​​the cylinder if its height is 3 cm.

Solution. It is necessary to use the formula for the total area of ​​a circular straight cylinder. But it lacks data, namely the radius of the base. But the area of ​​the circle is known. It is easy to calculate the radius from this.

It turns out to be equal to the square root of the quotient, which is obtained by dividing the area of ​​the base by pi. After dividing 12.56 by 3.14, the result is 4. The square root of 4 is 2. Therefore, the radius will have this value.

Answer: S floor = 50.24 cm 2.

Task No. 2. A cylinder with a radius of 5 cm is cut by a plane parallel to the axis. The distance from the section to the axis is 3 cm. The height of the cylinder is 4 cm. You need to find the cross-sectional area.

Solution. The cross-sectional shape is rectangular. One of its sides coincides with the height of the cylinder, and the other is equal to the chord. If the first quantity is known, then the second one needs to be found.

To do this, additional construction must be made. At the base we draw two segments. They will both start at the center of the circle. The first will end at the center of the chord and equal to the known distance to the axis. The second is at the end of the chord.

You will get a right triangle. The hypotenuse and one of the legs are known in it. The hypotenuse coincides with the radius. The second leg is equal to half the chord. The unknown leg multiplied by 2 will give the desired chord length. Let's calculate its value.

In order to find the unknown leg, you will need to square the hypotenuse and the known leg, subtract the second from the first and take the square root. The squares are 25 and 9. Their difference is 16. After taking the square root, 4 remains. This is the desired leg.

The chord will be equal to 4 * 2 = 8 (cm). Now you can calculate the cross-sectional area: 8 * 4 = 32 (cm 2).

Answer: S cross is equal to 32 cm 2.

Task No. 3. It is necessary to calculate the axial cross-sectional area of ​​the cylinder. It is known that a cube with an edge of 10 cm is inscribed in it.

Solution. The axial section of the cylinder coincides with a rectangle that passes through the four vertices of the cube and contains the diagonals of its bases. The side of the cube is the generatrix of the cylinder, and the diagonal of the base coincides with the diameter. The product of these two quantities will give the area that you need to find out in the problem.

To find the diameter, you will need to use the knowledge that the base of the cube is a square, and its diagonal forms an equilateral right triangle. Its hypotenuse is the desired diagonal of the figure.

To calculate it, you will need the formula of the Pythagorean theorem. You need to square the side of the cube, multiply it by 2 and take the square root. Ten to the second power is one hundred. Multiplied by 2 is two hundred. The square root of 200 is 10√2.

The section is again a rectangle with sides 10 and 10√2. Its area can be easily calculated by multiplying these values.

Answer. S section = 100√2 cm 2.

How to calculate the surface area of ​​a cylinder is the topic of this article. In any mathematical problem, you need to start by entering data, determine what is known and what to operate with in the future, and only then proceed directly to the calculation.

This volumetric body is a cylindrical geometric figure, bounded at the top and bottom by two parallel planes. If you apply a little imagination, you will notice that a geometric body is formed by rotating a rectangle around an axis, with one of its sides being the axis.

It follows that the curve described above and below the cylinder will be a circle, the main indicator of which is the radius or diameter.

Surface area of ​​a cylinder - online calculator

This function finally simplifies the calculation process, and it all comes down to automatically substituting the specified values ​​for the height and radius (diameter) of the base of the figure. The only thing that is required is to accurately determine the data and not make mistakes when entering numbers.

Cylinder side surface area

First you need to imagine what a scan looks like in two-dimensional space.

This is nothing more than a rectangle, one side of which is equal to the circumference. Its formula has been known since time immemorial - 2π *r, Where r- radius of the circle. The other side of the rectangle is equal to the height h. Finding what you are looking for will not be difficult.

Sside= 2π *r*h,

where is the number π = 3.14.

Total surface area of ​​a cylinder

To find the total area of ​​the cylinder, you need to use the resulting S side add the areas of two circles, the top and bottom of the cylinder, which are calculated using the formula S o =2π * r 2 .

The final formula looks like this:

Sfloor= 2π * r 2+ 2π * r * h.

Area of ​​a cylinder - formula through diameter

To facilitate calculations, it is sometimes necessary to perform calculations through the diameter. For example, there is a piece of hollow pipe of known diameter.

Without bothering ourselves with unnecessary calculations, we have a ready-made formula. 5th grade algebra comes to the rescue.

Sgender = 2π * r 2 + 2 π * r * h= 2 π * d 2 /4 + 2 π*h*d/2 = π *d 2 /2 + π *d*h,

Instead of r V full formula need to insert value r =d/2.

Examples of calculating the area of ​​a cylinder

Armed with knowledge, let's start practicing.

Example 1. It is necessary to calculate the area of ​​a truncated piece of pipe, that is, a cylinder.

We have r = 24 mm, h = 100 mm. You need to use the formula through the radius:

S floor = 2 * 3.14 * 24 2 + 2 * 3.14 * 24 * 100 = 3617.28 + 15072 = 18689.28 (mm 2).

We convert to the usual m2 and get 0.01868928, approximately 0.02 m2.

Example 2. It is required to find out the area of ​​the internal surface of an asbestos stove pipe, the walls of which are lined with refractory bricks.

The data is as follows: diameter 0.2 m; height 2 m. We use the formula in terms of diameter:

S floor = 3.14 * 0.2 2 /2 + 3.14 * 0.2 * 2 = 0.0628 + 1.256 = 1.3188 m2.

Example 3. How to find out how much material is needed to sew a bag, r = 1 m and 1 m high.

One moment, there is a formula:

S side = 2 * 3.14 * 1 * 1 = 6.28 m2.

Conclusion

At the end of the article, the question arose: are all these calculations and conversions of one value to another really necessary? Why is all this needed and most importantly, for whom? But don’t neglect and forget simple formulas from high school.

The world has stood and will stand on elementary knowledge, including mathematics. And, when starting any important work, it is never a bad idea to refresh your memory of these calculations, applying them in practice with great effect. Precision is the politeness of kings.

Cylinder radius formula:
where V is the volume of the cylinder, h is the height

A cylinder is a geometric body that is obtained by rotating a rectangle around its side. Also, a cylinder is a body bounded by a cylindrical surface and two parallel planes intersecting it. This surface is formed when a straight line moves parallel to itself. In this case, the selected point of the straight line moves along a certain plane curve (guide). This straight line is called the generator of the cylindrical surface.
Cylinder radius formula:
where Sb is the lateral surface area, h is the height

A cylinder is a geometric body that is obtained by rotating a rectangle around its side. Also, a cylinder is a body bounded by a cylindrical surface and two parallel planes intersecting it. This surface is formed when a straight line moves parallel to itself. In this case, the selected point of the straight line moves along a certain plane curve (guide). This straight line is called the generator of the cylindrical surface.
Cylinder radius formula:
where S is the total surface area, h is the height