How to find and what the circumference will be equal to. Calculating the radius: how to find the circumference of a circle knowing the diameter

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, is quite simple. We will look at all available methods 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:

• Consists of points A and B and all others from which AB can be seen at right angles. Diameter of this figure equal to length the segment under consideration.
• Includes only those 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 squares of the distances to the other two is a given value, which is always more 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 is further complicated by the fact that not everyone knows the basic geometric concepts. Radius is a segment that connects the center of a figure to a point on a curve. A special case in trigonometry is the unit circle. A chord is a segment that connects two points on a curve. For example, the already discussed AB falls under this definition. The diameter is the chord passing through the center. The number π is equal to the length of a unit semicircle.

Basic formulas

From the definitions it follows directly geometric formulas, which allow you to calculate the main characteristics of a 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 equal to 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 quotient of 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 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 the product of π and the square of the diameter divided 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 by diameter

For simplicity of explanation, let us denote by letters the characteristics of the figure necessary for the calculation. Let C be the desired length, D its diameter, and π approximately equal to 3.14. If we have only one known quantity, then the problem can be considered solved. Why is this necessary in life? Suppose we decide to surround a round pool with a fence. How to calculate required amount columns? And here the ability to calculate the circumference 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 required distance to the fence. For example, suppose that our home artificial pond is 20 meters wide, and we are going to place the posts at a ten-meter distance from it. The diameter of the resulting circle is 20 + 10*2 = 40 m. Length is 3.14*40 = 125.6 meters. We will need 25 posts if the gap between them is about 5 m.

Length through radius

As always, let's start by assigning letters to the characteristics of the circle. In fact, they are universal, so mathematicians from different countries It is not at all necessary to know each other's language. Let's assume that C is the circumference of the circle, r is its radius, and π is approximately equal to 3.14. The formula in this case looks like this: C = 2*π*r. Obviously, this is an absolutely correct equation. 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. To prevent it from getting dirty, we need a decorative wrapper. But how to cut a circle the right 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.

Sample problems

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

Example solution

First, let's write down what we are given: C = 26 cm, π = 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 actual 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 = 4.14 cm. It is important not to forget to write the answer correctly, that is, with units of measurement, otherwise the entire practical meaning of such problems is lost. In addition, for such inattention you can receive a grade one point lower. And no matter how annoying it may be, you will have to put up with this state of affairs.

The beast is not as scary as it is painted

So we have dealt with such a difficult task at first glance. As it turns out, you just need to understand the meaning of the terms and remember a few simple formulas. Math is not that scary, you just need to put in a little effort. So geometry is waiting for you!

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

The circle is drawn using 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. From the definition it follows that all radii of one circle are equal to each other.

A straight line segment (AB) connecting any two points of a 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 almost some cases, the circumference can be found by direct measurement. This can be done, for example, when measuring a circle comparatively small items(bucket, glass, etc.). To do this, you can use a tape measure, braid or cord.

In mathematics, the technique of indirectly determining the circumference is used. It consists of calculating using a 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 another is the length of the diameter). Naturally, for small objects these numbers will be small, and for large ones - 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. Let us denote the circumference of the circle by the letter WITH, length of diameter letter D, then their ratio will look like C:D. Actual measurements are always accompanied by inevitable inaccuracies. But, having completed the indicated experiment and made the necessary calculations, we get for the ratio C:D approximately the following numbers: 3.13; 3.14; 3.15. These numbers differ very little from one another.

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

Let's write a formula to determine the circumference.

Because C:D= π , That

C = πD

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

Task 1. Find the circumference ( WITH) of a round room if its diameter is 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. Let's find the wheel diameter:

125.6: 3.14 = 40 (cm).

Let us now 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. Firstly, 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 (a round flower bed, a pool, a fountain, etc.) with a sheet of paper. Thirdly, having counted the cells, we still do not receive any rule that allows us to solve another similar problem. Because of this, we will act differently. Let's compare the circle with some figure familiar to us and do it as follows: cut a circle out of paper, cut it in half first along the diameter, then cut each half in half, each quarter in half, etc., until we cut the circle, for example, into 32 parts shaped like teeth (Fig. 29).

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

The length of this figure (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 a letter WITH, radius letter r, then we can write the 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 multiplied by the radius.

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

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

2) Length of half circle C / 2 = 25.12: 2= 12.56 (cm).

3) Area of ​​the circle S = C / 2 r= 12.56 4 = 50.24 (sq. cm).

§ 118. Surface and volume of a cylinder.

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

The following have a cylinder shape (Fig. 31): a bucket, a glass (not faceted), a saucepan and many other objects.

Full surface cylinder (as well as the full surface rectangular parallelepiped) consists of the lateral surface and the areas of two bases (Fig. 32).

To clearly imagine 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, i.e. two circles, and cut the side surface lengthwise and unfold it, then it will be completely clear how to calculate the total surface of the cylinder. Side surface will unfold into a rectangle whose base is equal to the circumference. Therefore, the solution to the problem will look like:

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

2) Lateral surface area: 64.684 30.5 = 1972.862 (cm2).

3) Area of ​​one base: 32.342 10.3 = 333.1226 (sq.cm).

4) Full cylinder surface:

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

Task 2. Find volume iron barrel, having the shape of 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).

Our unit of volume measurement will be 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 laid on the base area, you need to calculate the base area of ​​the 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 has the form:

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

2) Area of ​​the circle: 94.2 30 = 2826 (sq. cm).

3) Cylinder volume: 2826,110 = 310,860 (cc. cm).

Answer. Barrel volume 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 to determine 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 problems, it is often necessary to calculate the circumference. Let's imagine a worker who produces round parts according to the diameters specified to him. Every time he knows the diameter, he must calculate the circumference. To save time and insure himself against mistakes, he turns to ready-made tables that indicate the diameters and the corresponding circumference lengths.

We will present a small part of such tables and tell you how to use them.

Let it be known that the diameter of the circle is 5 m. We look 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= 10 cm, then the circumference is 31.416 cm.

Using the same tables, you can also perform reverse calculations. If the circumference of a circle is known, then the corresponding diameter can be found in the table. Let the circumference be approximately 34.56 cm. Let us find in the table the number closest to this. This will be 34.558 (difference 0.002). The diameter corresponding to this 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 arithmetic problem book by S. A. Ponomarev and N. I. Sirneva.

A circle consists of many points that are at equal distances 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 what field he works in. Many vegetables and fruits, devices and mechanisms, dishes and furniture are round in shape. A circle is the set of points that lies within the boundaries of the 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. Interior 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 is not equal to unity, the ratio AX/BX. In a circle, this condition must be met; otherwise, this figure does not have the shape of a circle. Each point that makes up a figure is subject to the following rule: the sum of the squared distances from these points to the other two 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 relating to it. The main parameters of the figure are diameter, radius and chord. The radius is the segment connecting the center of the circle with any point on its curve. The magnitude of a chord is equal to the distance between two points on the curve of the 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 dimensions of a circle:

Diameter in calculation formulas

In economics and mathematics there is often a need to find the circumference of a circle. But also in Everyday life you may encounter this need, for example, when building a fence around a pool round shape. How to calculate the circumference of a circle by diameter? In this case, use the formula C = 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 required 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 of them will be needed.

Radius calculations

How to calculate the circumference of a circle from a known radius? To do this, use the formula C = 2*π*r, where C is the length, r is the radius. The radius in a circle is half the diameter, and this rule can be useful in everyday life. For example, in the case of preparing a pie in a sliding form.

To prevent the culinary product from getting dirty, it is necessary to use a decorative wrapper. How to cut a paper circle of the appropriate 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 shape is 20 centimeters, respectively, its radius is 10 centimeters. According to these parameters there is required size circle: 2*10*3, 14 = 62.8 centimeters.

Handy calculation methods

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

• At small sizes of a round object, its length can be found using a rope wrapped around it once.
• The size of a large object is measured as follows: a rope is laid out on a flat surface, and a circle is rolled along it once.
• Modern students and schoolchildren use calculators for calculations. Online, you can find out unknown quantities using known parameters.

Round objects in the history of human life

The first round-shaped product that man invented was the wheel. The first structures were small round logs mounted on an axle. 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 wheel upholstery that scientists of past centuries were looking for a formula for calculating this value.

A potter's wheel has the shape of a wheel, most parts in complex mechanisms, designs of water mills and spinning wheels. Round objects are often found in construction - frames of round windows in Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in their field professional activity are faced with the need to calculate the size of a circle.

Did you know that a person forgets about 40% information that he perceived. It follows from this that remembering everything, and especially knowing everything, is very difficult, and sometimes even unrealistic. For example, after a student has graduated from school and then from college, for example, humanitarian specialty, and not in a technical degree (construction or engineering department), it can be said with a high probability that he has long forgotten elementary mathematics.

Do you remember how to find the height of a trapezoid, how to find the derivative of a function, or how to construct a graph correctly? Surely not. It is rare that anyone will be able to complete such a task without additional help. Take, for example, a student who did not study geometry well at school and simply forgot how to find the perimeter of a circle. This article will be useful to those who wish to recall school curriculum mathematics. Often this need arises among parents, to whom schoolchildren turn for help with homework in geometry, as well as students who are currently studying the material.

Necessary:

- a circle whose perimeter needs to be found;
- school compass and ruler;
- a piece of paper and a pencil;
- calculator.

Instructions:

• Finding the perimeter of a circle is a similar task to calculating the circumference of a circle. First you need to measure it radius . To do this you need to use a compass. We place one of its legs in the center of the circle, and the second at any point on the circle. Since the circle is a collection of all equally distant points from the center, where exactly the second leg of the compass will be does not matter, since the distance will be the same everywhere.
• If you don’t have a compass at hand, you can find out circle diameter using a ruler. To do this, measure the length by placing a ruler so that it passes through the center of the circle. The distance we get will be diameter . It is equal to two radii, so the formula given a little further remains relevant.
• If center of the circle is not marked, then use a ruler to measure the most long distance from one point on the circle to another. With this method of calculation, the resulting perimeter of the circle will be an inaccurate number, since we could not determine the diameter quite accurately. We measure the resulting distance on a ruler by applying a compass to it. We write down the result on a piece of paper. This is the radius of our circle.
• To find out the perimeter of a circle, you need to use formula . It is very simple: the radius of our circle is multiplied by two, and then multiplied by Pi , which is constant and equals the value 3,14 . It was calculated by ancient mathematicians, and subsequent generations have been successfully using it in calculations for thousands of years, so there is no doubt about its correctness. After we carry out the calculations, we get the number that is what we are looking for.
• For large circles, the algorithm and instructions for measuring remain the same, only the ruler and compass are replaced with a construction tape, and special programs for calculations.

A circle is a curved line that encloses a circle. In geometry, shapes are flat, so the definition refers to a two-dimensional image. It is assumed that all points of this curve are located at an equal distance from the center of the circle.

The circle has several characteristics on the basis of which calculations related to this geometric figure are made. These include: diameter, radius, area and circumference. These characteristics are interrelated, that is, to calculate them, information about at least one of the components is sufficient. For example, knowing only the radius geometric figure Using the formula you can find the circumference, diameter, and area.

• The radius of a circle is the segment inside the circle connected to its center.
• A diameter is a segment inside a circle connecting its points and passing through the center. Essentially, the diameter is two radii. This is exactly what the formula for calculating it looks like: D=2r.
• There is one more component of a circle - a chord. This is a straight line that connects two points on a circle, but does not always pass through the center. So the chord that passes through it is also called the diameter.

How to find out the circumference? Let's find out now.

Circumference: formula

The Latin letter p was chosen to denote this characteristic. Archimedes also proved that the ratio of the circumference of a circle to its diameter is the same number for all circles: this is the number π, which is approximately equal to 3.14159. The formula for calculating π is: π = p/d. According to this formula, the value of p is equal to πd, that is, the circumference: p= πd. Since d (diameter) is equal to two radii, the same formula for the circumference can be written as p=2πr. Let's consider the application of the formula using simple problems as an example:

Problem 1

At the base of the Tsar Bell the diameter is 6.6 meters. What is the circumference of the base of the bell?

1. So, the formula for calculating the circle is p= πd
2. Substitute the existing value into the formula: p=3.14*6.6= 20.724

Answer: The circumference of the bell base is 20.7 meters.

Problem 2

The artificial satellite of the Earth rotates at a distance of 320 km from the planet. The radius of the Earth is 6370 km. What is the length of the satellite's circular orbit?

1. 1. Calculate the radius of the circular orbit of the Earth satellite: 6370+320=6690 (km)
2. 2.Calculate the length of the satellite’s circular orbit using the formula: P=2πr
3. 3.P=2*3.14*6690=42013.2

Answer: the length of the circular orbit of the Earth satellite is 42013.2 km.

Methods for measuring circumference

Calculating the circumference of a circle is not often used in practice. The reason for this is the approximate value of the number π. In everyday life, to find the length of a circle, they use special device– curvimeter. An arbitrary starting point is marked on the circle and the device is led from it strictly along the line until they reach this point again.

How to find the circumference of a circle? You just need to keep simple calculation formulas in your head.