Axial section formula. How to find the area of ​​a cylinder

A cylinder is a figure consisting of a cylindrical surface and two circles located in 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. Lateral surface formula given 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. Need to calculate required values and substitute it 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.

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

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

Here a segment of length h is its generator and height.

It can be seen that the cylinder consists of two identical grounds(ellipses in this case), which lie in parallel planes, and the lateral surface. The latter belongs to all points of the forming lines.

Before moving on to considering the axial section of cylinders, we will tell you what types of these figures there are.

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

Axial section of a straight cylinder

Axial is any section of the 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 half a cylinder, which is the 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.

Let us write the formulas for the axial cross-sectional area of ​​the 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 picture above shows a slanted cylinder made of paper. If you make its axial section, you will no longer get a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of the cross-section of a straight cylinder, is equal to the diameter d of the base, the other is the length of the forming segment. Let's denote it b.

To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. Another angle between them is needed. Let us assume that the acute angle between the guide and the base is α. This will also be the angle between the sides of the parallelogram. Then the formula for the axial cross-sectional area of ​​an 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 expressions without derivation that allow us to calculate the diagonals of a parallelogram using known sides and the acute angle 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 you how to use the knowledge gained to solve the following problem. Let us be given a round straight cylinder. It is known that the axial cross section of a cylinder is square. What is the area of ​​this section if the entire figure is 100 cm 2?

To calculate the required area, you need 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 half the height h. Taking this into account, 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 a square section is equal to the diameter of the base of the figure, the following formula will be valid to calculate its area S:

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.

How to calculate the surface area of ​​a cylinder is the topic of this article. At any math 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 geometric figure cylindrical in shape, bounded above and below by two parallel planes. If you apply a little imagination, you can see that geometric body is formed by rotating a rectangle around an axis, with the axis being one of its sides.

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. Need to know the area inner surface 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, starting some important work, it is never a bad idea to refresh your memory of these calculations by applying them in practice with great effect. Accuracy - the politeness of kings.

The area of ​​each base of the cylinder is π r 2, the area of ​​both bases will be 2π r 2 (fig.).

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 to be 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 areas of its two bases to the area of ​​the lateral surface of the cylinder, we obtain the area full surface cylinder

S full =2πRH + 2πR 2 = 2πR (H + R).

Volume of a straight cylinder

Theorem. Volume of a straight cylinder equal to the product area of ​​its base to height , i.e.

where Q is the area of ​​the base, and H is the height of the cylinder.

Since the area of ​​the base of the cylinder is Q, then 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 a sequence of prisms whose bases are the described and inscribed polygons discussed above, and whose side edges are parallel to the generatrix of the given cylinder and have length H. These prisms are circumscribed 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 = 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 as an example.

A cylinder has three surfaces: a top, a base, and a 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 * 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