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pipe perimeter. How to find and what will be the circumference of a circle

§ 117. Circumference and area of ​​a circle.

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 on 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 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 With, 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.

As 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 ( With) 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 With, 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.

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 With = 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. 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.

And what is its difference from the circle. Take a pen or colors and draw a regular circle on a piece of paper. Paint over the entire middle of the resulting figure with a blue pencil. The red outline denoting the boundaries of the figure is a circle. But the blue content inside it is the circle.

The dimensions of a circle and a circle are determined by the diameter. On the red line denoting the circle, mark two points so that they are mirror images of each other. Connect them with a line. The segment must pass through the point at the center of the circle. This segment, connecting the opposite parts of the circle, is called the diameter in geometry.

A segment that does not extend through the center of the circle, but merges with it at opposite ends, is called a chord. Therefore, the chord passing through the point of the center of the circle is its diameter.

The diameter is denoted by the Latin letter D. You can find the diameter of a circle by such values ​​as the area, length and radius of the circle.

The distance from the center point to the point plotted on the circle is called the radius and is denoted by the letter R. Knowing the value of the radius helps to calculate the diameter of the circle in one simple step:

For example, the radius is 7 cm. We multiply 7 cm by 2 and get a value equal to 14 cm. Answer: D of a given figure is 14 cm.

Sometimes it is necessary to determine the diameter of a circle only by its length. Here it is necessary to apply a special formula to help determine the Formula L \u003d 2 Pi * R, where 2 is a constant value (constant), and Pi \u003d 3.14. And since it is known that R \u003d D * 2, the formula can be represented in another way

This expression is also applicable as a formula for the diameter of a circle. Substituting the known values ​​in the problem, we solve the equation with one unknown. Let's say the length is 7 m. Therefore:

Answer: The diameter is 21.98 meters.

If the value of the area is known, then the diameter of the circle can also be determined. The formula that applies in this case looks like this:

D = 2 * (S / Pi) * (1 / 2)

S - in this case Let's say in the problem it is equal to 30 square meters. m. We get:

D=2*(30/3.14)*(1/2) D=9.55414

When the value indicated in the problem is equal to the volume (V) of the ball, the following formula for finding the diameter is applied: D = (6 V / Pi) * 1/3.

Sometimes you have to find the diameter of a circle inscribed in a triangle. To do this, by the formula we find the radius of the presented circle:

R = S / p (S is the area of ​​the given triangle and p is the perimeter divided by 2).

The result is doubled, given that D = 2 * R.

It is often necessary to find the diameter of a circle in everyday life. For example, when determining what is equivalent to its diameter. To do this, wrap the finger of the potential owner of the ring with a thread. Mark the points of contact between the two ends. Measure the length from point to point with a ruler. The resulting value is multiplied by 3.14, following the formula for determining the diameter with a known length. So, the statement that knowledge in geometry and algebra will not be useful in life does not always correspond to reality. And this is a serious reason to treat school subjects more responsibly.

Very often, when solving school assignments in physics or physics, the question arises - how to find the circumference of a circle, knowing the diameter? In fact, there are no difficulties in solving this problem, you just need to clearly understand what formulas, concepts and definitions are required for this.

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Basic concepts and definitions

  1. The radius is the line connecting the center of the circle and its arbitrary point. It is denoted by the Latin letter r.
  2. A chord is a line connecting two arbitrary points on a circle.
  3. Diameter is the line connecting two points of a circle and passing through its center. It is denoted by the Latin letter d.
  4. - this is a line consisting of all points that are at an equal distance from one chosen point, called its center. Its length will be denoted by the Latin letter l.

The area of ​​a circle is the entire area enclosed within a circle. It's measured in square units and is denoted by the Latin letter s.

Using our definitions, we conclude that the diameter of a circle is equal to its largest chord.

Attention! From the definition of what the radius of a circle is, you can find out what the diameter of a circle is. These are two radii laid out in opposite directions!

Circle diameter.

Finding the circumference of a circle and its area

If we are given the radius of a circle, then the diameter of the circle is described by the formula d = 2*r. Thus, to answer the question of how to find the diameter of a circle, knowing its radius, the last one is enough multiply by two.

The formula for the circumference of a circle, expressed in terms of its radius, is l \u003d 2 * P * r.

Attention! The Latin letter P (Pi) denotes the ratio of the circumference of a circle to its diameter, and this is a non-periodic decimal fraction. In school mathematics, it is considered to be a known tabular value equal to 3.14!

Now let's rewrite the previous formula to find the circumference of a circle in terms of its diameter, remembering what its difference is in relation to the radius. Get: l \u003d 2 * P * r \u003d 2 * r * P \u003d P * d.

From the course of mathematics it is known that the formula describing the area of ​​a circle has the form: s \u003d P * r ^ 2.

Now let's rewrite the previous formula to find the area of ​​a circle in terms of its diameter. We get

s = P*r^2 = P*d^2/4.

One of the most difficult tasks in this topic is determining the area of ​​a circle in terms of the circumference and vice versa. We use the fact that s = P*r^2 and l = 2*P*r. From here we get r = l/(2*П). We substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4P). The circumference of a circle is determined in exactly the same way in terms of the area of ​​a circle.

Determining Radius Length and Diameter

Important! First of all, we will learn how to measure the diameter. It's very simple - we draw any radius, extend it in the opposite direction until it intersects with the arc. We measure the resulting distance with a compass and with the help of any metric tool we find out what we are looking for!

Let's answer the question of how to find out the diameter of a circle, knowing its length. To do this, we express it from the formula l \u003d P * d. We get d = l/P.

We already know how to find its diameter from the circumference of a circle, and we will find the radius in the same way.

l \u003d 2 * P * r, hence r \u003d l / 2 * P. In general, to find out the radius, it must be expressed in terms of the diameter and vice versa.

Let now it is required to determine the diameter, knowing the area of ​​the circle. We use the fact that s \u003d P * d ^ 2/4. We express from here d. It turns out d^2 = 4*s/P. To determine the diameter itself, you need to extract square root of the right side. It turns out d \u003d 2 * sqrt (s / P).

Solution of typical tasks

  1. Learn how to find the diameter given the circumference of a circle. Let it be equal to 778.72 kilometers. Need to find d. d \u003d 778.72 / 3.14 \u003d 248 kilometers. Let's remember what the diameter is and immediately determine the radius, for this we divide the value d defined above in half. It turns out r=248/2=124 kilometers.
  2. Consider how to find the length of a given circle, knowing its radius. Let r have a value of 8 dm 7 cm. Let's translate all this into centimeters, then r will be equal to 87 centimeters. Let's use the formula to find the unknown length of a circle. Then our desired will be equal to l=2*3.14*87=546.36cm. Let's translate our obtained value into integers of metric values ​​l \u003d 546.36 cm \u003d 5 m 4 dm 6 cm 3.6 mm.
  3. Suppose we need to determine the area of ​​a given circle using the formula in terms of its known diameter. Let d = 815 meters. Recall the formula for finding the area of ​​a circle. Substituting the given values ​​here, we get s \u003d 3.14 * 815 ^ 2/4 \u003d 521416.625 sq. m.
  4. Now we will learn how to find the area of ​​a circle, knowing the length of its radius. Let the radius be 38 cm. We use the formula we know. Substitute here the value given to us by condition. You get the following: s \u003d 3.14 * 38 ^ 2 \u003d 4534.16 square meters. cm.
  5. The last task is to determine the area of ​​the circle from the known circumference. Let l = 47 meters. s \u003d 47 ^ 2 / (4P) \u003d 2209 / 12.56 \u003d 175.87 sq. m.

Circumference

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 of the problem, the question of which is how to find circumference, is simple enough. 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 circle diameter 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!