Calculation of diameter knowing the circumference. Calculating the radius: how to find the circumference of a circle knowing the diameter

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 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, 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 in memory big number numbers, 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, which will help 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 by simple formulas, including 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.

Its diameter. To do this, you just need to apply the formula for circumference. L = n DHere: L – circumference, n– number Pi, equal to 3.14, D – diameter of the circle. Rearrange the required value in the formula for the circumference of the circle left side and get: D = L/n

Let's look at a practical problem. Suppose you need to make a cover for a round country well, access to which in this moment No. No, and inappropriate weather. But do you have data on length its circumference. Let's assume this is 600 cm. We substitute the values ​​into the indicated formula: D = 600/3.14 = 191.08 cm. So, the diameter of your is 191 cm. 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.

Helpful advice

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

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

Instructions

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

Sources:

• circle diameter calculation

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

Instructions

Finally, circle can be divided by constructing segments. A segment is a part of a circle made up of a chord and an arc of a circle. In this case, a chord is a segment connecting any two points on a circle. Using segments circle can be divided into infinite set parts with or without a formation at its center.

Video on the topic

note

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

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

Instructions

Use Pi to find the length of a diameter given the known circumference. This constant expresses a constant relationship between these two parameters of the circle - regardless of the size of the circle, dividing its circumference by the length of its diameter always gives the same number. It follows from this 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 a diameter, accuracy to hundredths of a unit, that is, 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 precise definition, then the required number of signs for pi can be found, for example, at this link - http://www.math.com/tables/constants/pi.htm.

Given the known 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 right triangle, the legs of which form sides of 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 known parties: d=√(a² + b²).

Division into several equal parts- a common task. This is how you can build 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 designated center (if the center is not marked, you will have to find it in any way);
• - protractor;
• - compass with stylus;
• - pencil;
• - ruler.

Instructions

The easiest way to divide circle into equal parts - using a protractor. Dividing 360° into the required number of parts, you get the angle. Start at any point on the circle - the corresponding radius will be the zero mark. Starting from there, make marks on the protractor that correspond 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, made up of six equilateral triangles. Set the compass solution, equal to radius circles, and make notches with them, 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 is divided 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, install a compass solution, equal to a circle. Place the compass needle on one end of the diameter and 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 one. Its ends will become the remaining two vertices of the square inscribed in circle.

Using the method described above, you can find the middle of any segment. As a consequence, with this method you can double the number of equal parts into which you circle. Having found the midpoint of each side of the correct n- inscribed in circle, you can draw perpendiculars to them, find the point of their intersection with circle yu and thus construct the vertices of a regular 2n-gon. This procedure can be repeated as many times as you like. So, the square turns into, that - 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 make it possible to divide the circle into equal parts with high accuracy. When it is necessary to divide a circle into equal parts, use the table below. To do this, you need to multiply the diameter of the circle being divided by the coefficient given in the table: K x D.

Helpful advice

Dividing a circle into three, six and twelve equal parts. Two perpendicular axes are drawn, which, intersecting the circle at points 1,2,3,4, divide it into four equal parts; Using the well-known division technique right angle Using a compass or square, bisectors of right angles are constructed into two equal parts, which, intersecting with the circle at points 5, 6, 7, and 8, divide each fourth part of the circle in half.

When carrying out constructions of various geometric shapes sometimes it is necessary to determine their characteristics: length, width, height, and so on. If we're talking about about a circle or circle, you often have to determine its diameter. A diameter is a straight line segment that connects the two points furthest from each other located on a circle.

You will need

• - yardstick;
• - compass;
• - calculator.

No matter what sphere of the economy a person works, wittingly or unwittingly he uses mathematical knowledge accumulated over many centuries. We come across devices and mechanisms containing circles every day. Round shape has a wheel, pizza, many vegetables and fruits when cut form a circle, as well as plates, cups, and much more. However, not everyone knows how to correctly calculate the circumference.

To calculate the circumference of a circle, you must first remember what a circle is. This is the set of all points of the plane equidistant from this one. A circle is a geometric locus of points on a plane located inside a circle. From the above it follows that the perimeter of a circle and the circumference are one and the same.

Methods for finding the circumference of a circle

In addition to the mathematical method of finding the perimeter of a circle, there are also practical ones.

• Take a rope or cord and wrap it around once.
• Then measure the rope, the resulting number will be the circumference.
• Roll the round object once and count the length of the path. If the item is very small, you can wrap it with twine several times, then unwind the thread, measure and divide by the number of turns.
• Find the required value using the formula:

L = 2πr = πD ,

where L is the required length;

π – constant, approximately equal to 3.14 r – radius of the circle, the distance from its center to any point;

D is the diameter, it is equal to two radii.

Applying the formula to find the circumference of a circle

• Example 1. Treadmill passes around a circle with a radius of 47.8 meters. Find the length of this treadmill, taking π = 3.14.

L = 2πr =2*3.14*47.8 ≈ 300(m)

Answer: 300 meters

• Example 2. A bicycle wheel, having turned 10 times, has traveled 18.85 meters. Find the radius of the wheel.

18.85: 10 =1.885 (m) is the perimeter of the wheel.

1.885: π = 1.885: 3.1416 ≈ 0.6(m) – required diameter

Answer: wheel diameter 0.6 meters

The amazing number pi

Despite the apparent simplicity of the formula, for some reason it is difficult for many to remember it. Apparently, this is due to the fact that the formula contains an irrational number π, which is not present in the formulas for the area of ​​​​other figures, for example, a square, triangle or rhombus. You just need to remember that this is a constant, that is, a constant meaning the ratio of the circumference to the diameter. About 4 thousand years ago, people noticed that the ratio of the perimeter of a circle to its radius (or diameter) is the same for all circles.

The ancient Greeks approximated the number π with the fraction 22/7. For a long timeπ was calculated as the average between the lengths of inscribed and circumscribed polygons in a circle. In the third century AD, a Chinese mathematician performed a calculation for a 3072-gon and obtained an approximate value of π = 3.1416. It must be remembered that π is always constant for any circle. Its designation with the Greek letter π appeared in the 18th century. This is the first letter of the Greek words περιφέρεια - circle and περίμετρος - perimeter. In the eighteenth century, it was proven that this quantity is irrational, that is, it cannot be represented in the form m/n, where m is an integer and n is a natural number.

The circle occurs at 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 simpler.

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 that have 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 by 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.

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? 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: The volume of a cylinder is equal to the area of ​​the base multiplied by 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. 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 you know the circumference of a circle, you can find the corresponding diameter 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.