It often sounds like part of a plane that is bounded by a circle. The circumference of a circle is a flat closed curve. All points located 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 = 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).

\(r\) – radius.

\(d\) – diameter.

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

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

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

In the Cartesian coordinate system \(xOy\) we can also introduce the equation of any circle. Let us denote the center of the circle by the point \(X\) , which will have coordinates \((x_0,y_0)\) . Let the radius of this circle be equal to \(τ\) . Let's take an arbitrary point \(Y\) whose coordinates we denote by \((x,y)\) (Fig. 2).

Using the formula for the distance between two points in our given coordinate system, 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 the center we have chosen. That is, by definition 3, we obtain 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 (perimeter of a circle)

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 by \(C\) and \(C"\) , whose radii are equal to \(τ\) and \(τ"\) . We will inscribe regular \(n\)-gons into these circles, the perimeters of which are equal to \(ρ\) and \(ρ"\), the lengths of the sides are equal to \(α\) and \(α"\), respectively. As we know, the side of a regular \(n\) square 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 relation \(\frac(ρ)(ρ")=\frac(2τ)(2τ") \) will be true regardless of the number of sides of the 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 inscribed regular polygons (that is, \(n→∞\)), we obtain the equality:

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

From the last two equalities we obtain 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 double radius is always the same number, regardless of the choice of the circle and its parameters, that is

\(\frac(C)(2τ)=const \)

This constant should be called the number “pi” and denoted \(π\) . Approximately, this number will be equal to \(3.14\) (there is no exact value of this number, since it is an irrational number). Thus

\(\frac(C)(2τ)=π \)

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

\(C=2πτ\)

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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.

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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 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 in everyday life you may encounter this need, for example, when building a fence around a round pool. 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. Using these parameters, the required circle size is found: 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:

• If a round object is small, 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 the Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in their professional activities are faced with the need to calculate the dimensions of a circle.

A circle is found in everyday life no less often than a rectangle. And for many people, the problem of how to calculate the circumference is difficult. And all because it has no corners. If they were available, everything would become much easier.

What is a circle and where does it occur?

This flat figure represents 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 of a circle, except for people who are engineers and designers. They create designs for mechanisms that use, for example, gears, portholes and wheels. Architects create houses with round or arched windows.

Each of these and other cases requires its own precision. Moreover, it turns out to be impossible to calculate the circumference absolutely accurately. This is due to the infinity of the main number in the formula. "Pi" is still being refined. And the rounded value is most often used. The degree of accuracy is chosen to give the most correct answer.

Designations of quantities and formulas

Now it’s easy to answer the question of how to calculate the circumference of a circle by radius; for this you will need the following formula:

Since 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 = π * d.

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

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

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

Examples of tasks

Task one

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

Solution. Here it is not difficult to understand how to calculate the circumference. You just need to use the first formula. Since the radius is known, all you need to do is substitute the values ​​and calculate. 2 multiplied by a radius of 5 cm gives 10. All that remains is 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. It is necessary 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 into the product of 2 and π. It turns out that the product will give the result: 6.28. After division, the number left is: 200. This is the desired value.

Answer: r = 200 mm.

Task three

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

Solution. Similar to the previous problem, you will need to divide the known length by the value of π, rounded to the nearest hundredth. As a result of this action, the number 18 is obtained. The result is obtained.

Answer: d = 18 cm.

Problem four

Condition. The clock hands are 3 and 5 cm long. You need 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. You need to use it twice.

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

For the second answer, you need to multiply 2, π and 5. The product will give the 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 far does it run in one full revolution of the wheel?

Solution. This distance is equal to the circumference. Therefore, you need to use a suitable formula. Namely, multiply the value of π and 2 m. Calculations give the result: 6.28 m.

Answer: The squirrel runs 6.28 m.

Very often, when solving school assignments in physics or science, 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 imagine what formulas,concepts and definitions are required for this.

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Basic concepts and definitions

1. 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 lying 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. is a line consisting of all points located at equal distances from one selected point, called its center. We will denote its length by the Latin letter l.

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

Using our definitions, we come to the conclusion 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!

Diameter of a circle.

Finding the circumference and area of ​​a circle

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, has the form l = 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 a previously known tabular value equal to 3.14!

Now let's rewrite the previous formula to find the circumference of a circle through its diameter, remembering what its difference is in relation to the radius. It will turn out: l = 2*P*r = 2*r*P = P*d.

From the mathematics course we know that the formula describing the area of ​​a circle has the form: s = П*r^2.

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

s = П*r^2 = П*d^2/4.

One of the most difficult tasks in this topic is determining the area of ​​a circle through the circumference and vice versa. Let's take advantage of the fact that s = П*r^2 and l = 2*П*r. From here we get r = l/(2*П). Let's substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4П). In a completely similar way, the circumference is determined through the area of ​​the circle.

Determining radius length and diameter

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

Let us 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 = П*d. We get d = l/P.

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

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

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

Solving typical tasks

1. Let's find out how to find the diameter if the circumference is given. Let it be equal to 778.72 kilometers. Required to find d. d = 778.72/3.14 = 248 kilometers. Let's remember what a diameter is and immediately determine the radius; to do this, we divide the value d determined above in half. It will work out r = 248/2 = 124 kilometer
2. Let's 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 convert 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 value will be equal to l = 2*3.14*87 = 546.36 cm. Let's convert our obtained value into integer numbers of metric quantities l = 546.36 cm = 5 m 4 dm 6 cm 3.6 mm.
3. Let us need to determine the area of ​​a given circle using the formula through its known diameter. Let d = 815 meters. Let's remember the formula for finding the area of ​​a circle. Let's substitute the values ​​given to us here, we get s = 3.14*815^2/4 = 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 known to us. Let us substitute here the value given to us by condition. You get the following: s = 3.14*38^2 = 4534.16 sq. cm.
5. The last task is to determine the area of ​​a circle based on the known circumference. Let l = 47 meters. s = 47^2/(4P) = 2209/12.56 = 175.87 sq. m.

Circumference