S side surface of the cylinder. How to find the area of ​​a cylinder

The bodies of rotation studied in school are the cylinder, cone and ball.

If in a problem on the Unified State Exam in mathematics you need to calculate the volume of a cone or the area of ​​a sphere, consider yourself lucky.

Apply formulas for volume and surface area of ​​a cylinder, cone and sphere. All of them are in our table. Learn by heart. This is where knowledge of stereometry begins.

Sometimes it's good to draw the view from above. Or, as in this problem, from below.

2. How many times is the volume of a cone circumscribed about a regular quadrangular pyramid greater than the volume of a cone inscribed in this pyramid?

It's simple - draw the view from below. We see that the radius of the larger circle is times larger than the radius of the smaller one. The heights of both cones are the same. Therefore, the volume of the larger cone will be twice as large.

Another important point. Remember that in the problems of part B Unified State Exam options in mathematics, the answer is written as a whole number or a finite decimal fraction. Therefore, there should not be any or in your answer in part B. There is no need to substitute the approximate value of the number either! It must definitely shrink! It is for this purpose that in some problems the task is formulated, for example, as follows: “Find the area of ​​the lateral surface of the cylinder divided by.”

Where else are the formulas for volume and surface area of ​​bodies of revolution used? Of course, in problem C2 (16). We will also tell you about it.

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 in this case one can express the radius from the circumference of a circle, 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 usually does not translate into 3.14159... It just needs to be added next to numerical value, which was obtained as a result of 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.

When studying stereometry, one of the main topics is “Cylinder”. The area of ​​the lateral surface is considered, if not the main, then an important formula when solving geometric problems. However, it is important to remember the definitions that will help you navigate the examples and when proving various theorems.

Cylinder concept

First there are a few definitions to consider. Only after studying them can we begin to consider the question of the formula for the area of ​​the lateral surface of a cylinder. Based on this record, other expressions can be calculated.

• A cylindrical surface is understood as a plane described by a generatrix that moves and remains parallel to a given direction, sliding along an 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, the indicated geometric body is obtained.
• By axis we mean a straight line passing through both bases of the figure.
• A cylinder is a stereometric body bounded by an intersecting side surface and two parallel planes.

There are varieties of this volumetric figure:

1. By circular we mean a cylinder whose guide 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.
2. There is a straight cylinder. It received its name due to the perpendicularity of the forming figure to the bases.
3. The third type is a beveled cylinder. In textbooks you can find another name for it: “a circular cylinder with a beveled base.” This figure is determined by the radius of the base, minimum and maximum heights.
4. An equilateral cylinder is understood as a body having equal height and diameter of a circular plane.

Legend

Traditionally, the main “components” of the cylinder are called as follows:

• The radius of the base is R (it also replaces the similar value of a stereometric figure).
• Generator - L.
• Height - H.
• The area of ​​the base is S base (in other words, it is necessary to find the specified parameter of the circle).
• The heights of the beveled cylinder are 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 due to the fact that this formula is included in several other, more complex ones. Therefore, it is necessary to be well versed in theory.

The main components of the figure are:

1. Side surface. As is known, it is obtained due to the movement of the generatrix along a given curve.
2. The complete surface includes the existing bases and the side plane.
3. The cross-section of a 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 length and width are also components of other figures. So, conventionally, the lengths of the section are the generators. Width - parallel chords of a stereometric figure.
4. By axial section we mean the location of the plane through the center of the body.
5. And finally, a final definition. A tangent is a plane passing through the generatrix of the cylinder and located 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 expressions are given for a beveled cylinder, and then for a straight one.

Examples with a disassembled solution

It is necessary to find out the area of ​​the lateral surface of the cylinder. The diagonal of the section AC = 8 cm is given (and it is axial). Upon contact with the generatrix it turns out< ACD = 30°

Solution. Since the diagonal and angle values ​​are known, then in this case:

• CD = AC*cos 30°.

A comment. Triangle ACD, in the specific example, is rectangular. This means that the quotient of CD and AC = cosine of the existing angle. The meaning of trigonometric functions can be found in a special table.

Similarly, you can find the value of AD:

• AD = AC*sin 30°

Now you need to calculate the desired result using the following formulation: the area of ​​the 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 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.

Cylinders are given. Their volume = 128*p cm³. Which cylinder has the smallest total surface 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 “indicator” will be reached at the extremum point. To obtain the last value, you must use differentiation.

Formulas can be viewed in a special table for finding derivatives. Subsequently, the found result is equated to zero and a solution to the equation is found.

Answer: S min will be achieved 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 as VSKM, i.e., a rectangle, then its side BM = h. VMC needs to be considered. A triangle is a right triangle. 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 = BC = 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 a 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 - trapezoid. In this particular case, it is considered isosceles, since a circle is circumscribed around it.

If you find the height of the resulting trapezoid, 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 to consolidate the material

Given a cylinder. The lateral surface area is used in the subsequent solution. Other parameters are known. The base area is Q, the axial sectional area is M. It is necessary to find S. In other words, the total area of ​​the cylinder.

Given a cylinder. The area of ​​the lateral surface 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.

Surface area of ​​a cylinder. In this article we will look at tasks related to surface area. The blog has already covered tasks with a body of rotation such as a cone. A cylinder also belongs to bodies of revolution. What is required and needed to know about the surface area of ​​a cylinder? Let's look at the development of the cylinder:

The upper and lower base are two equal circles:

The side surface is a rectangle. Moreover, one side of this rectangle is equal to the height of the cylinder, and the other is equal to the circumference of the base. Let me remind you that the circumference of a circle is:

So, the formula for the surface of a cylinder is:

*No need to learn this formula! It is enough to know the formulas for the area of ​​a circle and the length of its circumference, then you can always write down the specified formula. Understanding it is important! Let's consider the tasks:

The circumference of the base of the cylinder is 3. The lateral surface area is 6. Find the height and surface area of ​​the cylinder (assume that Pi is 3.14 and round the result to the nearest tenth).

Total surface area of ​​the cylinder:

The circumference of the base and the lateral surface area of ​​the cylinder are given. That is, we are given the area of ​​a rectangle and one of its sides, we need to find the other side (this is the height of the cylinder):

The radius is required and then we can find the specified area.

The circumference of the base is equal to three, then we write:

Thus

Rounding to the nearest tenth, we get 7.4.

Answer: h = 2; S = 7.4

The lateral surface area of ​​the cylinder is 72Pi and the diameter of the base is 9. Find the height of the cylinder.

Means

Answer: 8

The lateral surface area of ​​the cylinder is 64Pi, and the height is 8. Find the diameter of the base.

The lateral surface area of ​​the cylinder is found by the formula:

The diameter is equal to two radii, which means:

Answer: 8

27058. The radius of the base of the cylinder is 2 and the height is 3. Find the lateral surface area of ​​the cylinder divided by Pi.

27133. The circumference of the base of the cylinder is 3, the height is 2. Find the area of ​​the lateral surface of the cylinder.

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 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. Accuracy - the politeness of kings.