How to determine the perimeter of a circle knowing the diameter. How to find and what will be the circumference of a circle

In whatever area of ​​the economy a person works, voluntarily or involuntarily, he uses mathematical knowledge accumulated over many centuries. We encounter devices and mechanisms containing circles every day. A round shape has a wheel, pizza, many vegetables and fruits in the section form a circle, as well as plates, cups, and much more. However, not everyone knows how to correctly calculate the circumference.

To calculate the circumference of a circle, you must first remember what a circle is. This is the set of all points in the plane equidistant from the given one. A circle is a locus of points in a plane that is inside a circle. From the above, it follows that the perimeter of a circle and the circumference of a circle are one and the same.

Ways to find the circumference of a circle

Apart from mathematical way finding the perimeter of a circle, there are also practical ones.

• Take a rope or cord and wrap it around once.
• Then measure the rope, the resulting number will be the circumference.
• Roll a round object once and calculate the length of the path. If the object is very small, you can wrap it with twine several times, then unwind the thread, measure and divide by the number of turns.
• Find the required value using the formula:

L = 2πr = πD ,

where L is the desired length;

π is a constant, approximately equal to 3.14 r is the radius of the circle, the distance from its center to any point;

D is the diameter, it is equal to two radii.

Applying the formula to find the circumference of a circle

• Example 1. The treadmill runs around a circle with a radius of 47.8 meters. Find the length of this treadmill, assuming π = 3.14.

L \u003d 2πr \u003d 2 * 3.14 * 47.8 ≈ 300 (m)

Answer: 300 meters

• Example 2. A bicycle wheel, turning around 10 times, traveled 18.85 meters. Find the radius of the wheel.

18.85: 10 = 1.885 (m) is the perimeter of the wheel.

1.885: π \u003d 1.885: 3.1416 ≈ 0.6 (m) - the desired diameter

Answer: wheel diameter 0.6 meters

The amazing number π

Despite the apparent simplicity of the formula, for some reason it is difficult for many to remember it. Apparently, this is due to the fact that the formula contains an irrational number π, which is not present in the area formulas of other figures, for example, a square, a triangle or a rhombus. You just need to remember that this is a constant, that is, a constant, meaning the ratio of the circumference to the diameter. About 4 thousand years ago, people noticed that the ratio of the perimeter of a circle to its radius (or diameter) is the same for any circles.

The ancient Greeks approximated the number π with the fraction 22/7. For a long timeπ was calculated as the average between the lengths of inscribed and circumscribed polygons in a circle. In the third century AD, a Chinese mathematician performed a calculation for a 3072-gon and obtained an approximate value of π = 3.1416. It must be remembered that π is always constant for any circle. Its designation with the Greek letter π appeared in the 18th century. This is the first letter Greek wordsπεριφέρεια - circumference and περίμετρος - perimeter. In the eighteenth century, it was proved that this quantity is irrational, that is, it cannot be represented as m/n, where m is an integer and n is a natural number.

Many objects in the environment have round shape. These are wheels, round window openings, pipes, various utensils and much more. You can calculate the circumference of a circle by knowing its diameter or radius.

There are several definitions of this geometric figure.

• It is a closed curve consisting of points that are located at the same distance from a given point.
• This is a curve consisting of points A and B, which are the ends of the segment, and all points from which A and B are visible at right angles. In this case, the segment AB is the diameter.
• For the same segment AB, this curve includes all points C such that the ratio AC/BC is constant and does not equal 1.
• This is a curve consisting of points for which the following is true: if you add the squares of the distances from one point to two given other points A and B, you get a constant number greater than 1/2 of the segment connecting A and B. This definition is derived from the Pythagorean theorem.

Note! There are other definitions as well. A circle is an area within a circle. The perimeter of a circle is its length. By different definitions the circle may or may not include the curve itself, which is its boundary.

Definition of a circle

Formulas

How to calculate the circumference of a circle using the radius? This is done with a simple formula:

where L is the desired value,

π is the number pi, approximately equal to 3.1413926.

Usually, to find the desired value, it is enough to use π up to the second decimal place, that is, 3.14, this will provide the desired accuracy. On calculators, in particular engineering ones, there may be a button that automatically enters the value of the number π.

Notation

To find through the diameter, there is the following formula:

If L is already known, you can easily find out the radius or diameter. To do this, L must be divided by 2π or π, respectively.

If a circle is already given, you need to understand how to find the circumference from this data. The area of ​​a circle is S = πR2. From here we find the radius: R = √(S/π). Then

L = 2πR = 2π√(S/π) = 2√(Sπ).

Calculating the area in terms of L is also easy: S = πR2 = π(L/(2π))2 = L2/(4π)

Summarizing, we can say that there are three main formulas:

• through the radius – L = 2πR;
• through the diameter - L = πD;
• through the area of ​​a circle – L = 2√(Sπ).

Pi

Without the number π, it will not be possible to solve the problem under consideration. The number π was found for the first time as the ratio of the circumference of a circle to its diameter. This was done by the ancient Babylonians, Egyptians and Indians. They found it quite accurately - their results differed from the now known value of π by no more than 1%. The constant was approximated by such fractions as 25/8, 256/81, 339/108.

Further, the value of this constant was considered not only from the point of view of geometry, but also from the point of view of mathematical analysis through the sums of the rows. The notation for this constant with the Greek letter π was first used by William Jones in 1706, and became popular after the work of Euler.

It is now known that this constant is an infinite non-periodic decimal, it is irrational, that is, it cannot be represented as a ratio of two integers. With the help of calculations on supercomputers in 2011, they learned the 10-trillion sign of a constant.

It is interesting! To memorize the first few characters of the number π, various mnemonic rules were invented. Some allow you to store a large number of digits in memory, for example, one French poem will help you remember pi up to 126 characters.

If you need the circumference, the online calculator will help you with this. There are many such calculators, they only need to enter the radius or diameter. Some of them have both of these options, others calculate the result only through R. Some calculators can calculate the desired value with different accuracy, you need to specify the number of decimal places. Also, using online calculators, you can calculate the area of ​​a circle.

Such calculators are easy to find with any search engine. There are also mobile applications, which will help solve the problem of how to find the circumference of a circle.

Useful video: circumference

Practical use

Solving such a problem is most often necessary for engineers and architects, but in everyday life, knowledge of the necessary formulas can also come in handy. For example, it is required to wrap a cake baked in a form with a diameter of 20 cm with a paper strip. Then it will not be difficult to find the length of this strip:

L \u003d πD \u003d 3.14 * 20 \u003d 62.8 cm.

Another example: you need to build a fence around a circular pool at a certain distance. If the radius of the pool is 10 m, and the fence needs to be placed at a distance of 3 m, then R for the resulting circle will be 13 m. Then its length is:

L \u003d 2πR \u003d 2 * 3.14 * 13 \u003d 81.68 m.

Useful video: circle - radius, diameter, circumference

Outcome

The perimeter of a circle is easy to calculate with simple formulas involving diameter or radius. You can also find the desired value through the area of ​​the circle. Online calculators or mobile applications will help to solve this problem, in which you need to enter singular is the diameter or radius.

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. 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 reservoir 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 it is not necessary 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 real ones. math problems 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!

Often sounds like a part of a plane that is bounded by a circle. The circumference of a circle is a flat closed curve. All points on the curve are the same distance from the center of the circle. In a circle, its length and perimeter are the same. The ratio of the length of any circle and its diameter is constant and is denoted by the number π \u003d 3.1415.

Determining the perimeter of a circle

The perimeter of a circle of radius r is equal to twice the product of radius r and the number π(~3.1415)

Circle Perimeter Formula

Perimeter of a circle of radius \(r\) :

\[ \LARGE(P) = 2 \cdot \pi \cdot r \]

\[ \LARGE(P) = \pi \cdot d \]

\(P \) - perimeter (circumference).

\(r\) is the radius.

\(d \) - diameter.

We will call such a circle geometric figure, which will consist of all such points that are at the same distance from any given point.

circle center we will call the point that is specified within the framework of Definition 1.

Circle radius we will call the distance from the center of this circle to any of its points.

AT Cartesian system coordinates \(xOy \) we can also enter the equation of any circle. Denote the center of the circle by a point \(X \) , which will have coordinates \((x_0,y_0) \) . Let the radius of this circle be \(τ \) . Take an arbitrary point \(Y \) , whose coordinates are denoted by \((x,y) \) (Fig. 2).

According to the formula for the distance between two points in the coordinate system we specified, we get:

\(|XY|=\sqrt((x-x_0)^2+(y-y_0)^2) \)

On the other hand, \(|XY| \) is the distance from any point on the circle to our chosen center. That is, by definition 3, we get that \(|XY|=τ \) , therefore

\(\sqrt((x-x_0)^2+(y-y_0)^2)=τ \)

\((x-x_0)^2+(y-y_0)^2=τ^2 \) (1)

Thus, we get that equation (1) is the equation of a circle in the Cartesian coordinate system.

Circumference (circle circumference)

We will derive the length of an arbitrary circle \(C \) using its radius equal to \(τ \) .

We will consider two arbitrary circles. Let us denote their lengths as \(C \) and \(C" \) , whose radii are \(τ \) and \(τ" \) . We will inscribe in these circles regular \(n\)-gons whose perimeters are equal to \(ρ \) and \(ρ" \) , whose side lengths are equal to \(α \) and \(α" \) , respectively. As we know, the side of a regular \(n\)-gon inscribed in a circle is equal to

\(α=2τsin\frac(180^0)(n) \)

Then, we will get that

\(ρ=nα=2nτ\frac(sin180^0)(n) \)

\(ρ"=nα"=2nτ"\frac(sin180^0)(n) \)

\(\frac(ρ)(ρ")=\frac(2nτsin\frac(180^0)(n))(2nτ"\frac(sin180^0)(n))=\frac(2τ)(2τ" )\)

We get that the ratio \(\frac(ρ)(ρ")=\frac(2τ)(2τ") \) will be true regardless of the value of the number of sides inscribed regular polygons. That is

\(\lim_(n\to\infty)(\frac(ρ)(ρ"))=\frac(2τ)(2τ") \)

On the other hand, if we infinitely increase the number of sides of the inscribed regular polygons (that is, \(n→∞ \) ), we will get the equality:

\(lim_(n\to\infty)(\frac(ρ)(ρ"))=\frac(C)(C") \)

From the last two equalities, we get that

\(\frac(C)(C")=\frac(2τ)(2τ") \)

\(\frac(C)(2τ)=\frac(C")(2τ") \)

We see that the ratio of the circumference of a circle to its doubled radius is always the same number, regardless of the choice of the circle and its parameters, that is

\(\frac(C)(2τ)=const \)

This constant is called the number "pi" and denoted \ (π \) . Approximately, this number will be equal to \ (3,14 \) (there is no exact value for this number, since it is an irrational number). In this way

\(\frac(C)(2τ)=π\)

Finally, we get that the circumference (perimeter of the circle) is determined by the formula

\(C=2πτ\)

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 problem 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, cut it first in diameter in half, 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 quite clear how to calculate the total surface of the cylinder. Side surface unfolds 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: cylinder volume equal to area base multiplied by height.

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

When solving various production tasks often need to calculate the circumference of a circle. 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.