Formulas for finding the diameter. How to calculate the circumference of a circle if the diameter and radius of the circle are not specified

1. Circumference. A circle is a closed flat curved line, all points of which are at an equal distance from one point (O), called the center of the circle (Fig. 27).

The circle is drawn with a compass. To do this, the sharp leg of the compass is placed in the center, and the other (with a pencil) is rotated around the first until the end of the pencil draws a complete circle. The distance from the center to any point on the circle is called its radius. It follows from the definition that all radii of one circle are equal to each other.

A straight line segment (AB) connecting any two points of the circle and passing through its center is called diameter. All diameters of one circle are equal to each other; the diameter is equal to two radii.

How to find the circumference of a circle? In practice, in some cases, the circumference can be found by direct measurement. This can be done, for example, when measuring the circumference of relatively small objects (bucket, glass, etc.). To do this, you can use a tape measure, braid or cord.

In mathematics, the method of indirectly determining the circumference of a circle is used. It consists in the calculation according to the ready-made formula, which we will now derive.

If we take several large and small round objects (coin, glass, bucket, barrel, etc.) and measure the circumference and diameter of each of them, we will get two numbers for each object (one measuring the circumference, and the other is the length of the diameter). Naturally, for small objects, these numbers will be small, and for large objects, they will be large.

However, if in each of these cases we take the ratio of the two numbers obtained (circumference and diameter), then with careful measurement we will find almost the same number. Denote the circumference by the letter FROM, the length of the diameter by the letter D, then their relation will look like C:D. Actual measurements are always accompanied by inevitable inaccuracies. But, having performed the indicated experiment and having made the necessary calculations, we will obtain for the relation C:D approximately the following numbers: 3.13; 3.14; 3.15. These numbers differ very little from each other.

In mathematics, by theoretical considerations, it is established that the desired ratio C:D never changes and it is equal to an infinite non-periodic fraction, the approximate value of which, with an accuracy of ten thousandths, is equal to 3,1416 . This means that any circle is longer than its diameter by the same number of times. This number is usually denoted by the Greek letter π (pi). Then the ratio of the circumference to the diameter is written as: C:D = π . We will limit this number only to hundredths, i.e., take π = 3,14.

Let's write a formula for determining the circumference of a circle.

Because C:D= π , then

C = πD

i.e. the circumference is equal to the product of the number π for diameter.

Task 1. Find the circumference ( FROM) of a round room if its diameter D= 5.5 m.

Taking into account the above, we must increase the diameter by 3.14 times to solve this problem:

5.5 3.14 = 17.27 (m).

Task 2. Find the radius of a wheel whose circumference is 125.6 cm.

This task is the reverse of the previous one. Find the wheel diameter:

125.6: 3.14 = 40 (cm).

Now let's find the radius of the wheel:

40: 2 = 20 (cm).

2. Area of ​​a circle. To determine the area of ​​a circle, one could draw a circle of a given radius on paper, cover it with transparent checkered paper, and then count the cells inside the circle (Fig. 28).

But this method is inconvenient for many reasons. First, near the contour of the circle, a number of incomplete cells are obtained, the size of which is difficult to judge. Secondly, you cannot cover a large object with a sheet of paper (a round flower bed, a pool, a fountain, etc.). Thirdly, having counted the cells, we still do not get any rule that allows us to solve another similar problem. Because of this, let's do it differently. Let's compare the circle with some figure familiar to us and do it as follows: cut out a circle from paper, first cut it in half in diameter, then cut each half in half again, each quarter in half again, etc., until we cut the circle, for example, into 32 parts having the shape of teeth (Fig. 29).

Then we fold them as shown in Figure 30, i.e., first we place 16 teeth in the form of a saw, and then we put 15 teeth into the holes formed, and finally, cut the last remaining tooth along the radius in half and attach one part to the left, the other - on right. Then you get a figure resembling a rectangle.

The length of this figure (the base) is approximately equal to the length of the semicircle, and the height is approximately equal to the radius. Then the area of ​​such a figure can be found by multiplying the numbers expressing the length of the semicircle and the length of the radius. If we denote the area of ​​a circle by the letter S, the circumference of the letter FROM, radius letter r, then we can write a formula for determining the area of ​​a circle:

which reads like this: The area of ​​a circle is equal to the length of the semicircle times the radius.

A task. Find the area of ​​a circle whose radius is 4 cm. First find the circumference, then the length of the semicircle, and then multiply it by the radius.

1) Circumference FROM = π D= 3.14 8 = 25.12 (cm).

2) Half circle length C / 2 \u003d 25.12: 2 \u003d 12.56 (cm).

3) Circle area S = C / 2 r\u003d 12.56 4 \u003d 50.24 (sq. cm).

§ 118. Surface and volume of a cylinder.

Task 1. Find the total surface area of ​​a cylinder with a base diameter of 20.6 cm and a height of 30.5 cm.

The shape of a cylinder (Fig. 31) is: a bucket, a glass (not faceted), a saucepan and many other items.

The full surface of a cylinder (like the full surface of a rectangular parallelepiped) consists of the side surface and the areas of the two bases (Fig. 32).

To visualize what we are talking about, you need to carefully make a model of a cylinder out of paper. If we subtract two bases from this model, that is, two circles, and cut the lateral surface lengthwise and unfold it, then it will be completely clear how to calculate the total surface of the cylinder. The side surface will unfold into a rectangle, the base of which is equal to the circumference of the circle. Therefore, the solution to the problem will look like:

1) Circumference: 20.6 3.14 = 64.684 (cm).

2) Side surface area: 64.684 30.5= 1972.862(sq.cm).

3) The area of ​​one base: 32.342 10.3 \u003d 333.1226 (sq. cm).

4) Full surface of the cylinder:

1972.862 + 333.1226 + 333.1226 = 2639.1072 (sq cm) ≈ 2639 (sq cm).

Task 2. Find the volume of an iron barrel shaped like a cylinder with dimensions: base diameter 60 cm and height 110 cm.

To calculate the volume of a cylinder, you need to remember how we calculated the volume of a rectangular parallelepiped (it is useful to read § 61).

The unit of measure for volume is the cubic centimeter. First you need to find out how many cubic centimeters can be placed on the base area, and then multiply the found number by the height.

To find out how many cubic centimeters can be placed on the base area, you need to calculate the base area of ​​\u200b\u200bthe cylinder. Since the base is a circle, you need to find the area of ​​the circle. Then, to determine the volume, multiply it by the height. The solution to the problem looks like:

1) Circumference: 60 3.14 = 188.4 (cm).

2) Area of ​​a circle: 94.230 = 2826 (sq. cm).

3) Cylinder volume: 2826 110 \u003d 310 860 (cc).

Answer. The volume of the barrel is 310.86 cubic meters. dm.

If we denote the volume of a cylinder by the letter V, base area S, cylinder height H, then you can write a formula for determining the volume of a cylinder:

V = S H

which reads like this: The volume of a cylinder is equal to the area of ​​the base times the height.

§ 119. Tables for calculating the circumference of a circle by diameter.

When solving various production problems, it is often necessary to calculate the circumference. Imagine a worker who manufactures round parts according to the diameters indicated to him. He must each time, knowing the diameter, calculate the circumference. To save time and insure himself against mistakes, he turns to ready-made tables that indicate the diameters and the corresponding circumferences.

Here is a small part of these tables and tell you how to use them.

Let it be known that the diameter of the circle is 5 m. We are looking for in the table in the vertical column under the letter D number 5. This is the length of the diameter. Next to this number (to the right, in the column called "Circumference") we will see the number 15.708 (m). In exactly the same way, we find that if D\u003d 10 cm, then the circumference is 31.416 cm.

The same tables can be used to perform reverse calculations. If the circumference is known, then you can find the corresponding diameter in the table. Let the circumference be approximately 34.56 cm. Let's find in the table the number closest to the given one. This will be 34.558 (0.002 difference). The diameter corresponding to such a circumference is approximately 11 cm.

The tables mentioned here are available in various reference books. In particular, they can be found in the book "Four-digit mathematical tables" by V. M. Bradis. and in the problem book on arithmetic by S. A. Ponomarev and N. I. Syrnev.

A circle is found in everyday life no less than a rectangle. And for many people, the task of how to calculate the circumference of a circle is difficult. And all because she has no corners. With them, everything would be much easier.

What is a circle and where does it occur?

This flat figure is a number of points that are located at the same distance from another one, which is the center. This distance is called the radius.

In everyday life, it is not often necessary to calculate the circumference, except for people who are engineers and designers. They design mechanisms that use, for example, gears, portholes and wheels. Architects create houses that have round or arched windows.

Each of these and other cases requires its own precision. Moreover, it is absolutely impossible to calculate the circumference of a circle with absolute accuracy. This is due to the infinity of the main number in the formula. "Pi" is still being specified. And most often the rounded value is used. The degree of accuracy is chosen so as to give the most correct answer.

Notation of quantities and formulas

Now it is easy to answer the question of how to calculate the circumference of a circle from a radius, this will require the following formula:

Since the radius and diameter are related to each other, there is another formula for calculations. Since the radius is two times smaller, the expression will change slightly. And the formula for how to calculate the circumference of a circle, knowing the diameter, will be as follows:

l \u003d π * d.

What if you need to calculate the perimeter of a circle?

Just remember that a circle includes all points inside the circle. So, its perimeter coincides with its length. And after calculating the circumference, put an equal sign with the perimeter of the circle.

By the way, they have the same designations. This applies to the radius and diameter, and the Latin letter P is the perimeter.

Task examples

Task one

Condition. Find the circumference of a circle whose radius is 5 cm.

Solution. Here it is easy to understand how to calculate the circumference of a circle. You just need to use the first formula. Since the radius is known, you only need to substitute the values ​​​​and count. 2 multiplied by a radius of 5 cm gives 10. It remains to multiply it by the value of π. 3.14 * 10 = 31.4 (cm).

Answer: l = 31.4 cm.

Task two

Condition. There is a wheel whose circumference is known and equal to 1256 mm. You need to calculate its radius.

Solution. In this task, you will need to use the same formula. But only the known length will need to be divided by the product of 2 and π. It turns out that the product will give the result: 6.28. After division, the number remains: 200. This is the desired value.

Answer: r = 200 mm.

Task three

Condition. Calculate the diameter if the circumference is known, which is 56.52 cm.

Solution. Similar to the previous problem, you need to divide the known length by the value of π, rounded up to hundredths. As a result of such an action, the number 18 is obtained. The result is obtained.

Answer: d = 18 cm.

Task four

Condition. The clock hands are 3 and 5 cm long. It is necessary to calculate the lengths of the circles that describe their ends.

Solution. Since the arrows coincide with the radii of the circles, the first formula is required. It needs to be used twice.

For the first length, the product will consist of factors: 2; 3.14 and 3. The result will be the number 18.84 cm.

For the second answer, you need to multiply 2, π and 5. The product will give a number: 31.4 cm.

Answer: l 1 = 18.84 cm, l 2 = 31.4 cm.

Task five

Condition. A squirrel runs in a wheel with a diameter of 2 m. How much distance does it run in one complete revolution of the wheel?

Solution. This distance is equal to the circumference of the circle. Therefore, you need to use the appropriate formula. Namely, multiply the value of π and 2 m. The calculations give the result: 6.28 m.

Answer: Squirrel runs 6.28 m.

We are surrounded by many things. And many of them are round. It is given to them for easy use. Take, for example, a wheel. If it were made in the shape of a square, how would it roll along the road?

In order to make a round object, you need to know what the formula for the circumference of a circle looks like through the diameter. To do this, we first define what this concept is.

Circle and circumference

A circle is a set of points that are placed at an equal distance from the main point - the center. This distance is called the radius.

The distance between two points on a given line is called a chord. In addition, if the chord passes through the main point (center), then it is called a diameter.

Now consider what a circle is. The collection of all points that are inside the outline is called a circle.

What is the circumference of a circle?

After we have considered all the definitions, we can calculate the diameter of the circle. The formula will be discussed a little later.

To begin with, we will try to measure the length of the outline of the glass. To do this, we wrap it with a thread, then measure it with a ruler and find out the approximate length of the imaginary line around the glass. Because the size depends on the correct measurement of the item, and this method is not reliable. Nevertheless, it is quite possible to make accurate measurements.

To do this, again remember the wheel. We have repeatedly seen that if the spoke in the wheel (radius) is increased, then the length of the wheel rim (circumference) will also increase. And just as the radius of the circle decreases, the length of the rim also decreases.

If we carefully follow these changes, we will see that the length of an imaginary circular line is proportional to its radius. And this number is constant. Next, consider how the diameter of a circle is determined: the formula for this will be applied in the example below. Let's take a look at it step by step.

Circle formula in terms of diameter

Since the length of the outline is proportional to the radius, it is also proportional to the diameter. Therefore, we will conditionally denote its length by the letter C, diameter - d. Since the ratio of outline length to diameter is a constant number, it can be determined.

Having done all the calculations, we will determine a number that is approximately equal to 3.1415 ... For the reason that the calculations did not work out a specific number, we will denote it by the letter π . This icon is useful to us in order to derive the formula for the circumference of a circle through the diameter.

Let's draw an imaginary line through the central point and measure the distance between the two extreme ones. This will be the diameter. If we know the diameter of a circle, the formula for determining its length itself will look like this: C=d*pi.

If we determine the length of different outlines, then if their diameter is known, the same formula will be applied. Since the sign π - this is an approximate calculation, then it was decided to multiply the diameter by 3.14 (a number rounded to hundredths).

How to calculate diameter: formula

This time, let's try using this formula to calculate other values, in addition to the length of the outline. To calculate the diameter from the circumference, the same formula is used. Only for this we divide its length by π . It will look like this d=C/π.

Let's see how this formula works in practice. For example, we know the length of the outline of the well, we should calculate its diameter. It is impossible to measure it, because due to weather conditions there is no access to it. And our task is to make a cover. What will we do in this case?

You need to use the formula. Let's take the length of the outline of the well - for example, 600 cm. We put a specific number in the formula, namely C \u003d 600 / 3.14. As a result, we will get approximately 191 cm. Round the result to 200 cm. Then, using a compass, draw a round line with a radius of 100 cm.

Since an outline with a large diameter must be drawn with an appropriate compass, such a tool can be made by yourself. To do this, take a rail of the desired length and drive in a nail at each end. We install one nail into the workpiece and drive it in lightly so that it does not move from the intended place. And with the help of the second we draw a line. The device is very simple and convenient.

Modern technologies allow you to use an online calculator to calculate the length of the outline. To do this, you just need to enter the diameter of the circle. The formula will be applied automatically. You can also calculate the circumference of a circle using the radius. In addition, if you know the circumference of a circle, the online calculator calculates the radius and diameter using this formula.

A circle is made up of many points that are equidistant from the center. This is a flat geometric figure, and finding its length is not difficult. A person encounters a circle and a circle every day, regardless of the area in which he works. Many vegetables and fruits, devices and mechanisms, dishes and furniture have a round shape. A circle is a set of points that is within the boundaries of a circle. Therefore, the length of the figure is equal to the perimeter of the circle.

Characteristics of the figure

In addition to the fact that the description of the concept of a circle is quite simple, its characteristics are also easy to understand. With their help, you can calculate its length. The inner part of the circle consists of many points, among which two - A and B - can be seen at right angles. This segment is called the diameter, it consists of two radii.

Within the circle there are points X such, which does not change and does not equal unity, the ratio AX / BX. In a circle, this condition is necessarily met, otherwise this figure does not have the shape of a circle. The rule applies to each point that makes up the figure: the sum of the squared distances from these points to two others always exceeds half the length of the segment between them.

Basic circle terms

In order to be able to find the length of a figure, you need to know the basic terms related to it. The main parameters of the figure are diameter, radius and chord. A radius is a segment that connects the center of a circle with any point on its curve. The value of a chord is equal to the distance between two points on the curved figure. Diameter - distance between points passing through the center of the figure.

Basic formulas for calculations

The parameters are used in the formulas for calculating the values ​​of the circle:

Diameter in calculation formulas

In economics and mathematics, it often becomes necessary to find the circumference of a circle. But in everyday life, you can also encounter this need, for example, during the construction of a fence around a round pool. How to calculate the circumference of a circle from a diameter? In this case, use the formula C \u003d π * D, where C is the desired value, D is the diameter.

For example, the width of the pool is 30 meters, and the fence posts are planned to be placed at a distance of ten meters from it. In this case, the formula for calculating the diameter is: 30+10*2 = 50 meters. The desired value (in this example, the length of the fence): 3.14 * 50 = 157 meters. If the fence posts stand at a distance of three meters from each other, then a total of 52 will be needed.

Radius calculations

How to calculate the circumference of a circle from a known radius? For this, the formula C \u003d 2 * π * r is used, where C is the length, r is the radius. The radius in a circle is less than half the diameter, and this rule can come in handy in everyday life. For example, in the case of making a pie in a sliding form.

In order for the culinary product not to get dirty, it is necessary to use a decorative wrapper. And how to cut a paper circle of a suitable size?

Those who are a little familiar with mathematics understand that in this case you need to multiply the number π by twice the radius of the shape used. For example, the diameter of the mold is 20 centimeters, respectively, its radius is 10 centimeters. According to these parameters, the required circle size is found: 2 * 10 * 3, 14 \u003d 62.8 centimeters.

Handy calculation methods

If it is not possible to find the circumference using the formula, then you should use the available methods for calculating this value:

• With a small round object, its length can be found using a rope wrapped around once.
• The size of a large object is measured as follows: a rope is laid out on a flat plane, and a circle is rolled over it once.
• Modern students and schoolchildren use calculators for calculations. Known parameters can be used to find out unknown values ​​online.

Round objects in the history of human life

The first round product that man invented was the wheel. The first structures were small rounded logs mounted on axles. Then came wheels made of wooden spokes and rims. Gradually, metal parts were added to the product to reduce wear. It was in order to find out the length of the metal strips for the upholstery of the wheel that scientists of past centuries were looking for a formula for calculating this value.

The potter's wheel has the shape of a wheel, most of the details in complex mechanisms, designs of water mills and spinning wheels. Often there are round objects in construction - the frames of round windows in the Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers daily in the field of their professional activities are faced with the need to calculate the size of a circle.

A circle is a closed curve, all points of which are at the same distance from the center. This figure is flat. Therefore, the solution to the problem, the question of which is how to find the circumference of a circle, is quite simple. All available methods, we will consider in today's article.

Figure descriptions

In addition to a fairly simple descriptive definition, there are three more mathematical characteristics of a circle, which in themselves contain the answer to the question of how to find the circumference of a circle:

• Consists of points A and B and all others from which AB can be seen at right angles. The diameter of this figure is equal to the length of the segment under consideration.
• Includes only points X such that the ratio AX/BX is constant and not equal to one. If this condition is not met, then it is not a circle.
• It consists of points, for each of which the following equality holds: the sum of the squared distances to the other two is a given value, which is always greater than half the length of the segment between them.

Terminology

Not everyone at school had a good math teacher. Therefore, the answer to the question of how to find the circumference of a circle is also complicated by the fact that not everyone knows the basic geometric concepts. Radius - a segment that connects the center of the figure with a point on the curve. A special case in trigonometry is the unit circle. A chord is a line segment that connects two points on a curve. For example, the already considered AB falls under this definition. Diameter is a chord passing through the center. The number π is equal to the length of the unit semicircle.

Basic formulas

Geometric formulas directly follow from the definitions, which allow you to calculate the main characteristics of the circle:

1. The length is equal to the product of the number π and the diameter. The formula is usually written as follows: C = π*D.
2. The radius is half the diameter. It can also be calculated by calculating the quotient of dividing the circumference by twice the number π. The formula looks like this: R = C/(2* π) = D/2.
3. The diameter is equal to the circumference divided by π or twice the radius. The formula is quite simple and looks like this: D = C/π = 2*R.
4. The area of ​​a circle is equal to the product of the number π and the square of the radius. Similarly, diameter can be used in this formula. In this case, the area will be equal to the quotient of dividing the product of the number π and the square of the diameter by four. The formula can be written as follows: S = π*R 2 = π*D 2 /4.

How to find the circumference of a circle from a diameter

For simplicity of explanation, we denote by letters the characteristics of the figure necessary for calculating. Let C be the desired length, D be its diameter, and let pi be approximately 3.14. If we have only one known quantity, then the problem can be considered solved. Why is it necessary in life? Suppose we decide to enclose a round pool with a fence. How to calculate the required number of columns? And here the ability to calculate the circumference of a circle comes to the rescue. The formula is as follows: C = π D. In our example, the diameter is determined based on the radius of the pool and the required distance to the fence. For example, suppose that our home artificial pond is 20 meters wide, and we are going to put posts at a distance of ten meters from it. The diameter of the resulting circle is 20 + 10 * 2 = 40 m. The length is 3.14 * 40 = 125.6 meters. We will need 25 columns if the gap between them is about 5 m.

Length through radius

As always, let's start by assigning letter circles to characteristics. In fact, they are universal, so mathematicians from different countries do not need to know each other's language. Suppose C is the circumference of a circle, r is its radius, and π is approximately 3.14. The formula looks like this in this case: C = 2*π*r. Obviously, this is an absolutely correct equality. As we have already figured out, the diameter of a circle is equal to twice its radius, so this formula looks like this. In life, this method can also often come in handy. For example, we bake a cake in a special sliding form. So that it does not get dirty, we need a decorative wrapper. But how to cut a circle of the desired size. This is where mathematics comes to the rescue. Those who know how to find out the circumference of a circle will immediately say that you need to multiply the number π by twice the radius of the shape. If its radius is 25 cm, then the length will be 157 centimeters.

Task examples

We have already considered several practical cases of the acquired knowledge on how to find out the circumference of a circle. But often we are not concerned with them, but with the real mathematical problems that are contained in the textbook. After all, the teacher gives points for them! Therefore, let's consider a problem of increased complexity. Let's assume that the circumference is 26 cm. How to find the radius of such a figure?

Example Solution

To begin with, let's write down what is given to us: C \u003d 26 cm, π \u003d 3.14. Also remember the formula: C = 2* π*R. From it you can extract the radius of the circle. Thus, R= C/2/π. Now let's proceed to the direct calculation. First, divide the length by two. We get 13. Now we need to divide by the value of the number π: 13 / 3.14 \u003d 4.14 cm. It is important not to forget to write down the answer correctly, that is, with units of measurement, otherwise the whole practical meaning of such problems is lost. In addition, for such inattention, you can get a score of one point lower. And no matter how annoying it may be, you have to put up with this state of affairs.

The beast is not as scary as it is painted

So we figured out such a difficult task at first glance. As it turned out, you just need to understand the meaning of the terms and remember a few easy formulas. Math is not so scary, you just need to make a little effort. So geometry is waiting for you!