What is the area of \u200b\u200bthe lateral surface of the cylinder. Axial section of a cylinder of direct and inclined

There are a large number of tasks related to the cylinder. They 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 \u200b\u200bthe cylinder and its volume.

Which body is a cylinder?

In the course of the school curriculum, a circular cylinder is studied, that is, being such at the base. But they also distinguish the elliptical appearance of this figure. 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 cylinder generators. All generators are parallel to each other and equal. They constitute the lateral surface of the body.

In the general case, a cylinder is an inclined body. If the generators make a right angle with the bases, then they are already talking about a straight figure.

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

The main elements of the cylinder

The main elements of the cylinder are as follows.

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. Matches the one that can be held 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 the cylinder crosses a plane containing an axis.
5. Tangent plane. It passes through one of the generators and is perpendicular to the axial section, which is drawn through this generatrix.

How is the cylinder connected with a prism inscribed in it or described near it?

Sometimes there are problems in which it is necessary to calculate the area of \u200b\u200bthe cylinder, and some elements of the prism connected with it are known. How do these figures relate?

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

The described prism has regular polygons in the bases. They are described near the circles of the cylinder, which are its bases. The planes that contain the faces of the prism touch the cylinder along generatrixes.

On the area of \u200b\u200bthe lateral surface and base for a straight circular cylinder

If you scan the side surface, you get a rectangle. Its sides will coincide with the generatrix and the circumference of the base. Therefore, the lateral area of \u200b\u200bthe cylinder will be equal to the product of these two quantities. If you write down the formula, you get the following:

S side \u003d l * n,

where n is the generator, l is the circumference.

Moreover, the last parameter is calculated by the formula:

l \u003d 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 \u003d π * r 2.

On the area of \u200b\u200bthe entire surface of a straight circular cylinder

Since it is formed by two bases and a lateral surface, it is necessary to add these three values. That is, the total area of \u200b\u200bthe cylinder will be calculated by the formula:

S gender \u003d 2 π * r * n + 2 π * r 2.

Often it is written in another form:

S gender \u003d 2 π * r (n + r).

On the areas of an inclined circular cylinder

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

To calculate the area of \u200b\u200bthe lateral surface of the inclined cylinder, it will be necessary to multiply the values \u200b\u200bof the generatrix and the perimeter of the section, which will be perpendicular to the selected generatrix.

The formula is as follows:

S side \u003d x * P,

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

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

l \u003d π * (a + c),

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 surface area must be calculated using the following expression:

S gender \u003d 2 π * r 2 + x * R.

What are some sections of a straight circular cylinder equal to?

When the cross section passes through the axis, its area is defined as the product of the generatrix and the diameter of the base. This is because it has the appearance of a rectangle, the sides of which coincide with the designated elements.

To find the cross-sectional area of \u200b\u200bthe cylinder, which is parallel to the axial, you will also need the formula for the rectangle. In this situation, one side will still coincide with the height, and the other is equal to the base chord. The latter coincides with the section line at the base.

When the cross section is perpendicular to the axis, then it has the form of a circle. Moreover, its area is the same as at the base of the figure.

Perhaps another intersection at some angle to the axis. Then in the cross section an oval or part thereof is obtained.

Examples of tasks

Task number 1.   A straight cylinder is given, the base area of \u200b\u200bwhich is 12.56 cm 2. It is necessary to calculate the total area of \u200b\u200bthe cylinder if its height is 3 cm.

Decision. You must use the formula for the full area of \u200b\u200ba circular straight cylinder. But it lacks data, namely the radius of the base. But the area of \u200b\u200bthe circle is known. It is easy to calculate the radius from it.

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

Answer: S floor \u003d 50.24 cm 2.

Task number 2.   A cylinder with a radius of 5 cm is nipped by a plane parallel to the axis. The distance from the cross section to the axis is 3 cm. The height of the cylinder is 4 cm. It is required to find the cross-sectional area.

Decision. The sectional shape is rectangular. One side coincides with the height of the cylinder, and the other is equal to the chord. If the first quantity is known, then the second must be found.

For this, an additional construction should be done. At the base, we draw two segments. Both of them will begin in the center of the circle. The first will end in the center of the chord and equal the known distance to the axis. The second is at the end of the chord.

The result is a right triangle. The hypotenuse and one of the legs are known in it. Hypotenuse coincides with the radius. The second leg is equal to half the chord. An unknown leg, multiplied by 2, will give the desired chord length. We calculate its value.

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

The chord will be equal to 4 * 2 \u003d 8 (cm). Now you can calculate the cross-sectional area: 8 * 4 \u003d 32 (cm 2).

Answer: S section is 32 cm 2.

Task number 3.   It is necessary to calculate the axial sectional area of \u200b\u200bthe cylinder. It is known that a cube with an edge of 10 cm is inscribed in it.

Decision. The axial section of the cylinder coincides with the 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 needs to be found in the problem.

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

To calculate it, you need the formula of the Pythagorean theorem. You need to square the side of the cube, multiply it by 2 and extract the square root. Ten in the second degree is a hundred. Multiplied by 2 - two hundred. The square root of 200 is 10√2.

The section is again a rectangle with sides 10 and 10√2. Its area is easy to calculate by multiplying these values.

Answer. S section \u003d 100√2 cm 2.

A cylinder is a symmetrical spatial figure whose properties are considered in high school with a course on stereometry. For its description, such linear characteristics as the height and radius of the base are used. In this article, we will consider questions regarding what an axial section of a cylinder is and how to calculate its parameters through the basic linear characteristics of a figure.

Geometric figure

First, we give a definition to the figure, which will be discussed in the article. A cylinder is a surface formed by parallel movement 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 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 last belongs to all the points of the generating lines.

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

If the generatrix is \u200b\u200bperpendicular to the base of the figure, then we speak of a straight cylinder. Otherwise, the cylinder will be inclined. If you connect the center points of two bases, then the resulting line is called the axis of the figure. The figure below shows the difference between straight and tilted cylinders.

It can be seen that for a straight figure, the length of the generatrix segment coincides with the value of height h. For an inclined cylinder, the height, that is, the distance between the bases, is always less than the length of the generatrix line.

Axial section of a straight cylinder

Axial 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 the cylinder, the straight axis passes through the center of the circle and is perpendicular to its plane. This means that the circle under consideration will intersect in its diameter. The figure shows the 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 straight circular cylinder is a rectangle. Its sides are the diameter d of the base and the height h of the figure.

We write the formulas for the axial sectional area of \u200b\u200bthe cylinder and the length h d of its diagonal:

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

To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. An angle between them is also needed. 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 axial sectional area of \u200b\u200bthe 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 present without derivation the expressions that allow us to calculate the diagonals of a parallelogram along the known sides and the acute angle between them:

l 1 \u003d √ (d 2 + b 2 - 2 * b * d * cos (α));

l 2 \u003d √ (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.

Direct cylinder problem

We 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 \u200b\u200bthis section equal if the whole figure is 100 cm 2?

To calculate the required area, it is necessary to 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 two times less than the height h. Given this, we can rewrite the equality above in the form:

S f \u003d 2 * pi * r * (r + 2 * r) \u003d 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, the following formula will be valid for calculating its area S:

S \u003d (2 * r) 2 \u003d 4 * r 2 \u003d 2 * S f / (3 * pi)

We see that the desired area is uniquely determined by the surface area of \u200b\u200bthe cylinder. Substituting the data into equality, we come to the answer: S \u003d 21.23 cm 2.

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 \u200b\u200bthe cylinder and, applying the formula, we will solve several problems as an example.

  A cylinder has three surfaces: the top, bottom, and side surface.

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

It is known that the area of \u200b\u200ba circle is πr 2. Therefore, the area formula of two circles (top and bottom of the cylinder) will have the form πr 2 + πr 2 \u003d 2πr 2.

  The third, lateral surface of the cylinder, is the curved wall of the cylinder. In order to better imagine this surface, we will try to transform it to get a recognizable shape. Imagine that a cylinder is an ordinary tin can that has no top cover and bottom. Make a vertical incision on the side wall from the top to the base of the can (Step 1 in the figure) and try to maximize open (straighten) the resulting figure (Step 2).

After full disclosure of the obtained jar, we will see a familiar figure (Step 3), this is a rectangle. The area of \u200b\u200bthe rectangle is easy to calculate. But before that, let's return for a moment to the original cylinder. The top of the original cylinder is a circle, and we know that the length of the circle is calculated by the formula: L \u003d 2πr. In the figure, it is marked in red.

When the side wall of the cylinder is fully open, we see that the circumference becomes the length of the resulting rectangle. The sides of this rectangle are the circumference (L \u003d 2πr) and the height of the cylinder (h). The area of \u200b\u200bthe rectangle is equal to the product of its sides - S \u003d length x width \u003d L x h \u003d 2πr x h \u003d 2πrh. As a result, we obtained a formula for calculating the area of \u200b\u200bthe lateral surface of the cylinder.

Formula of cylinder lateral surface area
S side. \u003d 2πrh

The total surface area of \u200b\u200bthe cylinder

Finally, if we add up the area of \u200b\u200ball three surfaces, we get the formula for the area of \u200b\u200bthe total surface of the cylinder. The surface area of \u200b\u200bthe cylinder is equal to the area of \u200b\u200bthe top of the cylinder + the area of \u200b\u200bthe base of the cylinder + the area of \u200b\u200bthe lateral surface of the cylinder or S \u003d πr 2 + πr 2 + 2πrh \u003d 2πr 2 + 2πrh. Sometimes this expression is written in the identical formula 2πr (r + h).

The formula for the total surface area of \u200b\u200ba cylinder
S \u003d 2πr 2 + 2πrh \u003d 2πr (r + h)
r is the radius of the cylinder, h is the height of the cylinder

Examples of calculating cylinder surface area

To understand the above formulas, let's try to calculate the surface area of \u200b\u200bthe cylinder using examples.

1. The radius of the base of the cylinder is 2, the height is 3. Determine the area of \u200b\u200bthe side surface of the cylinder.

The total surface area is calculated by the formula: S side. \u003d 2πrh

S side. \u003d 2 * 3.14 * 2 * 3

S side. \u003d 6.28 * 6

S side. \u003d 37.68

The area of \u200b\u200bthe lateral surface of the cylinder is 37.68.

2. How to find the surface area of \u200b\u200ba cylinder if the height is 4 and the radius is 6?

The total surface area is calculated by the formula: S \u003d 2πr 2 + 2πrh

S \u003d 2 * 3.14 * 6 2 + 2 * 3.14 * 6 * 4

S \u003d 2 * 3.14 * 36 + 2 * 3.14 * 24

A cylinder (comes from the Greek language, from the words “skating rink”, “roller”) is a geometric body that is bounded on the outside by a surface called a cylindrical 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 makes a movement along a curve of a flat type. This straight line is called the generatrix, and the curve line is called the guide.

The cylinder consists of a pair of bases and a lateral cylindrical surface. Cylinders come in several forms:

1. A circular, straight cylinder. In such a cylinder, the base and the guide are perpendicular to the generatrix line, and there is

2. Inclined cylinder. The angle between the generatrix line and the base is not straight.

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

The area of \u200b\u200bthe cylinder, as well as the total surface area of \u200b\u200bany cylinder, is found by adding together the base areas of this figure and the lateral surface area.

The formula by which the total cylinder area is calculated for a circular, straight cylinder:

Sp \u003d 2n Rh + 2n R2 \u003d 2n R (h + R).

The lateral surface area is found a little more complicated than the entire cylinder area, it is calculated by multiplying the length of the generatrix line by the perimeter of the section formed by a plane that is perpendicular to the generatrix line.

This cylinder for a circular, straight cylinder is recognized by the scan of this object.

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

It follows that the lateral area of \u200b\u200bthe cylinder is equal to the area of \u200b\u200bthe sweep and can be calculated by this formula:

If you take a circular, straight cylinder, then for him:

P \u003d 2n R, and Sb \u003d 2n Rh.

If the cylinder is inclined, then the area of \u200b\u200bthe lateral surface should be equal to the product of the length of its generatrix line and the perimeter of the section, which is perpendicular to this generatrix line.

Unfortunately, there is no simple formula for expressing the lateral surface area of \u200b\u200ban inclined cylinder through its height and its base parameters.

To calculate a cylinder, you need to know a few facts. If a section intersects the bases with its plane, 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 the diameter of the base of the cylinder. And the area of \u200b\u200bsuch a section, respectively, is equal to the product of one side of the rectangle on 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 base 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 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. Intersection points connect to the center. And the base of the triangle is the one sought after, it sounds like this: “The sum of the squares of two legs is equal to the hypotenuse squared”:

C2 \u003d 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 \u200b\u200ba circle.

The circumference is:

S okr. \u003d 2n R2.

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

R \u003d C \\ 2n, where n is the number pi, the mathematical constant calculated for working with circle data and equal to 3.14.

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 cylinder surface area formula includes a separate calculation of the base area and side surface. Since the bases in the cylinder are equal, its total area will be calculated by the formula:

An example of calculating the area of \u200b\u200ba cylinder we will consider after we learn all the necessary formulas. First, we need a formula for the area of \u200b\u200bthe base of the cylinder. Since the base of the cylinder is a circle, we need to apply:
  We remember that in these calculations a constant number Π \u003d 3.1415926 is used, which is calculated as the ratio of the circumference of a circle to its diameter. This number is a mathematical constant. An example of calculating the area of \u200b\u200bthe base of the cylinder we will also consider a little later.

Cylinder lateral surface area

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

Now consider the problem in which we need to calculate the total area of \u200b\u200bthe cylinder. In the given figure, the height is h \u003d 4 cm, r \u003d 2 cm. We find the total area of \u200b\u200bthe cylinder.
  To begin, calculate the area of \u200b\u200bthe bases:
  Now consider an example of calculating the lateral surface area of \u200b\u200ba cylinder. In expanded form, it represents a rectangle. Its area is calculated according to the above formula. Substitute all the data in it:
  The total area of \u200b\u200bthe circle is the sum of the double area of \u200b\u200bthe base and the side:

  Thus, using the formulas of the area of \u200b\u200bthe bases and the lateral surface of the figure, we were able to find the total surface area of \u200b\u200bthe 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 axial sectional area of \u200b\u200bthe cylinder is derived from the calculation formula: