How to find out the circumference knowing the diameter formula. How to find and what will be the circumference of a circle?

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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. The diameter of this figure is equal to the length of 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

The definitions directly follow geometric formulas that 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 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 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 the required distance from the fence. For example, let's assume 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 do not necessarily need to know each other’s languages. 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 of the required 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 a more complex problem. 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 get 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!

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 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

\(C=2πτ\)

And how is it different from a 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 indicating the boundaries of the shape 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 indicating the circle, mark two points so that they are mirror images of each other. Connect them with a line. The segment will definitely pass through the point in the center of the circle. This segment connecting opposite parts of a circle is called a diameter in geometry.

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

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

The distance from the central 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 the given figure is 14 cm.

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

This expression is also applicable as a formula for the diameter of a circle. Substituting the quantities known 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 area is known, then the diameter of the circle can also be determined. The formula that is used in in this case, looks like that:

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 used: D = (6 V / Pi) * 1 / 3.

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

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

We double the result obtained, taking into account that D = 2 * R.

Often you have to find the diameter of a circle in everyday life. For example, when determining what is equivalent to its diameter. To do this, you need to wrap the finger of the potential owner of the ring with thread. Mark the points of contact of the two ends. Measure the length from point to point with a ruler. We multiply the resulting value by 3.14, following the formula for determining the diameter with a known length. So, the statement that knowledge of geometry and algebra is not useful in life is not always true. And this is a serious reason for taking school subjects more responsibly.

Many objects in the world around us are round in shape. These are wheels, round window openings, pipes, various dishes and much more. You can calculate the length of a circle by knowing its diameter or radius.

There are several definitions of this geometric figure.

• This 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 not equal to 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. A circle is an area within a circle. The perimeter of a circle is its length. According to different definitions, a 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 using a simple formula:

where L is the desired value,

π is the number pi, approximately equal to 3.1413926.

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

Designations

To find through the diameter there is the following formula:

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

If a circle has already been given, you need to understand how to find the circumference from this data. The area of ​​the 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)

To summarize, we can say that there are three basic formulas:

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

Pi

Without the number π it will not be possible to solve the problem under consideration. The number π was first found 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 currently 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 calculated not only from the point of view of geometry, but also from the point of view of mathematical analysis through sums of series. The designation of this constant by the Greek letter π was first used by William Jones in 1706, and it became popular after the work of Euler.

It is now known that this constant is an infinite non-periodic decimal fraction; it is irrational, that is, it cannot be represented as a ratio of two integers. Using supercomputer calculations, the 10-trillionth sign of the constant was discovered in 2011.

This is interesting! Various mnemonic rules have been invented to remember the first few digits of the number π. Some allow you to store a large number of numbers in memory, for example, one French poem will help you remember pi up to the 126th digit.

If you need the circumference, an online calculator will help you with this. There are many such calculators; you just 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 precision, you need to specify the number of decimal places. You can also calculate the area of ​​a circle using online calculators.

Such calculators are easy to find with any search engine. There are also mobile applications that will help you 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 be useful. For example, you need to wrap a paper strip around a cake baked in a mold with a diameter of 20 cm. Then it will not be difficult to find the length of this strip:

L = πD = 3.14 * 20 = 62.8 cm.

Another example: you need to build a fence around a round 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 = 2πR = 2 * 3.14 * 13 = 81.68 m.

Useful video: circle - radius, diameter, circumference

Bottom line

The perimeter of a circle can be easily calculated using simple formulas involving diameter or radius. You can also find the desired quantity through the area of ​​a circle. Online calculators or mobile applications, in which you need to enter a single number - diameter or radius, will help you solve this problem.