What is the circumference of its circumference. How to find and what will be the circumference of a circle

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

The radius is the line connecting the center of the circle and its arbitrary point. It is denoted by the Latin letter r.
A chord is a line connecting two arbitrary points on a circle.
Diameter is the line connecting two points of a circle and passing through its center. It is denoted by the Latin letter d.
- 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

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.
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.
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.
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.
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

1. Harder to find circumference through diameter So let's take a look at this option first.

Example: Find the circumference of a circle whose diameter is 6 cm. We use the above formula for the circumference of a circle, but first we need to find the radius. To do this, we divide the diameter of 6 cm by 2 and get the radius of the circle 3 cm.

After that, everything is extremely simple: We multiply the number Pi by 2 and by the resulting radius of 3 cm.
2*3.14*3cm=6.28*3cm=18.84cm.

2. And now let's take a look at the simple option again find the circumference of a circle with a radius of 5 cm

Solution: The radius of 5 cm is multiplied by 2 and multiplied by 3.14. Do not be alarmed, because rearranging the factors does not affect the result, and circumference formula can be applied in any order.

5cm * 2 * 3.14 = 10 cm * 3.14 = 31.4 cm - this is the found circumference for a radius of 5 cm!

Online circumference calculator

Our circumference calculator will perform all these non-tricky calculations instantly and write the solution in a line with comments. We will calculate the circumference for a radius of 3, 5, 6, 8 or 1 cm, or the diameter is 4, 10, 15, 20 dm, our calculator does not care for which value of the radius to find the circumference.

All calculations will be accurate, tested by mathematicians. The results can be used in solving school problems in geometry or mathematics, as well as in working calculations in construction or in the repair and decoration of premises, when accurate calculations are required using this formula.

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.

A circle will be called such a 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.

In the Cartesian coordinate system \(xOy \) 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 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 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

\(C=2πτ \)

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Its diameter. To do this, you just need to apply the formula for the circumference of a circle. L \u003d p DHere: L - circumference, p- the number Pi, equal to 3.14, D - the diameter of the circle. Rearrange the formula for the circumference of the circle to the left side and get: D \u003d L / n

Let's analyze a practical problem. Suppose you need to make a cover for a round country well, to which there is currently no access. No, and unsuitable weather conditions. But do you have data on length its circumference. Suppose it is 600 cm. We substitute the values \u200b\u200bin the indicated formula: D \u003d 600 / 3.14 \u003d 191.08 cm. So, 191 cm is your diameter. Increase the diameter to 2, taking into account the allowance for the edges. Set the compass to a radius of 1 m (100 cm) and draw a circle.

Useful advice

It is convenient to draw circles of relatively large diameters at home with a compass, which can be quickly made. It is done like this. Two nails are driven into the rail at a distance from each other equal to the radius of the circle. Drive one nail shallowly into the workpiece. And use the other, rotating the rail, as a marker.

A circle is a geometric figure on a plane, which consists of all points of this plane that are at the same distance from a given point. The given point is called the center. circles, and the distance at which the points circles are from its center - radius circles. The area of the plane bounded by a circle is called a circle. There are several calculation methods diameter circles, the choice of a specific envy from the available initial data.

Instruction

In the simplest case, if a circle of radius R, then it will be equal to
D=2*R
If the radius circles is not known, but it is known, then the diameter can be calculated using the length formula circles
D = L/P, where L is the length circles, P - P.
Same diameter circles can be calculated, knowing the area bounded by it
D \u003d 2 * v (S / P), where S is the area of \u200b\u200bthe circle, P is the number of P.

Sources:

circle diameter calculation

In the course of high school planimetry, the concept circle is defined as a geometric figure consisting of all points of a plane lying at a radius distance from a point called its center. Inside the circle, you can draw many segments connecting its points in various ways. Depending on the construction of these segments, circle can be divided into several parts in different ways.

Instruction

Finally, circle can be divided into segments. A segment is a part of a circle made up of a chord and an arc of a circle. A chord in this case is a line segment joining any two points on the circle. Using segments circle can be divided into an infinite number of parts with or without education in its center.

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note

The figures obtained by the listed methods - polygons, segments and sectors, can also be divided using appropriate methods, for example, polygon diagonals or angle bisectors.

A circle is called a flat geometric figure, and the line that limits it is usually called a circle. The main property is that each point on this line is the same distance from the center of the figure. A segment starting at the center of the circle and ending at any of the points on the circle is called the radius, and a segment connecting two points of the circle and passing through the center is called the diameter.

Instruction

Use pi to find the length of a diameter given the circumference of a circle. This constant expresses a constant ratio between these two parameters of the circle - regardless of the size of the circle, dividing its circumference by the length of the diameter always gives the same number. From this it follows that to find the length of the diameter, the circumference should be divided by the number Pi. As a rule, for practical calculations of the length of the diameter, accuracy up to hundredths of a unit, that is, up to two decimal places, is sufficient, so the number Pi can be considered equal to 3.14. But since this constant is an irrational number, it has an infinite number of decimal places. If there is a need for more exact definition, then the required number of characters for pi can be found, for example, at this link - http://www.math.com/tables/constants/pi.htm.

Given the lengths of the sides (a and b) of a rectangle inscribed in a circle, the length of the diameter (d) can be calculated by finding the length of the diagonal of this rectangle. Since the diagonal here is the hypotenuse in a right-angled triangle, the legs of which form sides of a known length, then according to the Pythagorean theorem, the length of the diagonal, and with it the length of the diameter of the circumscribed circle, can be calculated by finding from the sum of the squares of the lengths of the known sides: d = √ (a² + b²).

Dividing into several equal parts is a common task. So you can build a regular polygon, draw a star, or prepare the basis for a diagram. There are several ways to solve this interesting problem.

You will need

- a circle with a marked center (if the center is not marked, you will have to find it in any way);
- protractor;
- compasses with lead;
- pencil;
- ruler.

Instruction

The easiest way to share circle into equal parts - with the help of a protractor. By dividing 360° into the required number of parts, you get the angle. Start at any point on the circle - the radius corresponding to it will be the zero mark. Starting from there, make marks on the protractor corresponding to the calculated angle. This method is recommended if you need to divide circle by five, seven, nine, etc. parts. For example, to build a regular pentagon, its vertices must be located every 360/5 = 72°, that is, at 0°, 72°, 144°, 216°, 288°.

To share circle into six parts, you can use the property of a regular one - its longest diagonal is equal to twice the side. A regular hexagon is, as it were, composed of six equilateral triangles. Set the compass opening equal to the radius of the circle, and make serifs with it, starting from any arbitrary point. The serifs form a regular hexagon, one of the vertices of which will be at this point. By connecting the vertices through one, you will build a regular triangle inscribed in circle, that is, it into three equal parts.

To share circle into four parts, start with an arbitrary diameter. Its ends will give two of the required four points. To find the rest, set the compass opening equal to the circle. Putting the compass needle on one of the ends of the diameter, make notches outside the circle and below. Repeat the same with the other end of the diameter. Draw an auxiliary line between the intersection points of the serifs. It will give you a second diameter strictly perpendicular to the original. Its ends will become the other two vertices of the square inscribed in circle.

Using the method described above, you can find the midpoint of any segment. As a consequence, this method can double the number of equal parts that you circle. Finding the midpoint of each side of a regular n- inscribed in circle, you can draw perpendiculars to them, find their point of intersection with circle yu and thus construct the vertices of a regular 2n-gon. This procedure can be repeated any time. So, the square turns into , that one - into, etc. Starting with a square, you can, for example, divide circle into 256 equal parts.

note

To divide a circle into equal parts, dividing heads or dividing tables are usually used, which allow dividing a circle into equal parts with high accuracy. When it is necessary to divide the circle into equal parts, use the table below. To do this, multiply the diameter of the divisible circle by the coefficient given in the table: K x D.

Useful advice

Division of a circle into three, six and twelve equal parts. Two perpendicular axes are drawn, which, crossing the circle at points 1,2,3,4, divide it into four equal parts; Using the well-known method of dividing a right angle into two equal parts using a compass or a square, they build right angle bisectors that intersect with the circle at points 5, 6, 7, and 8 divide each fourth part of the circle in half.

When building various geometric shapes sometimes you need to determine their characteristics: length, width, height, and so on. If we are talking about a circle or a circle, then it is often necessary to determine their diameter. Diameter is a line segment that connects two points on a circle that are farthest from each other.

You will need

- yardstick;
- compass;
- calculator.

The circle calculator is a service specially designed to calculate the geometric dimensions of figures online. Thanks to this service, you can easily determine any parameter of a figure based on a circle. For example: You know the volume of a sphere, but you need to get its area. There is nothing easier! Select the appropriate option, enter a numeric value, and click the Calculate button. The service not only displays the results of calculations, but also provides the formulas by which they were made. Using our service, you can easily calculate the radius, diameter, circumference (perimeter of a circle), the area of a circle and a ball, and the volume of a ball.

Calculate Radius

The task of calculating the value of the radius is one of the most common. The reason for this is quite simple, because knowing this parameter, you can easily determine the value of any other parameter of a circle or ball. Our site is built exactly on such a scheme. Regardless of which initial parameter you choose, the radius value is calculated first and all subsequent calculations are based on it. For greater accuracy of calculations, the site uses the number Pi rounded to the 10th decimal place.

Calculate Diameter

Diameter calculation is the simplest type of calculation that our calculator can perform. Getting the diameter value is not difficult at all and manually, for this you do not need to resort to the help of the Internet at all. The diameter is equal to the value of the radius multiplied by 2. The diameter is the most important parameter of the circle, which is extremely often used in everyday life. Absolutely everyone should be able to calculate it correctly and use it. Using the capabilities of our site, you will calculate the diameter with great accuracy in a fraction of a second.

Find out the circumference of a circle

You can't even imagine how many round objects around us and what an important role they play in our lives. The ability to calculate the circumference is necessary for everyone, from an ordinary driver to a leading design engineer. The formula for calculating the circumference is very simple: D=2Pr. The calculation can be easily carried out both on a piece of paper and with the help of this Internet assistant. The advantage of the latter is that it will illustrate all the calculations with drawings. And to everything else, the second method is much faster.

Calculate the area of a circle

The area of the circle - like all the parameters listed in this article, is the basis of modern civilization. To be able to calculate and know the area of a circle is useful for all segments of the population without exception. It is difficult to imagine an area of science and technology in which it would not be necessary to know the area of a circle. The formula for calculation is again not difficult: S=PR 2 . This formula and our online calculator will help you find the area of any circle effortlessly. Our site guarantees high accuracy of calculations and their lightning-fast execution.

Calculate the area of a sphere

The formula for calculating the area of a ball is no more complicated than the formulas described in the previous paragraphs. S=4Pr 2 . This simple set of letters and numbers has been giving people the ability to accurately calculate the area of a sphere for many years. Where can it be applied? Yes, everywhere! For example, you know that the area of the globe is 510,100,000 square kilometers. It is useless to list where knowledge of this formula can be applied. The scope of the formula for calculating the area of a ball is too wide.

Calculate the volume of a sphere

To calculate the volume of the ball, use the formula V=4/3(Pr 3). It was used to create our online service. The site site makes it possible to calculate the volume of a ball in a matter of seconds, if you know any of the following parameters: radius, diameter, circumference, area of a circle or area of a ball. You can also use it for inverse calculations, for example, to know the volume of a ball, get the value of its radius or diameter. Thank you for briefly reviewing the capabilities of our lap calculator. We hope you enjoyed your stay with us and have already added the site to your bookmarks.