How to calculate the diameter from a circle knowing the length. How to find and what will be the circumference of a circle

A circle is found in everyday life no less than a rectangle. And for many people, the task of how to calculate the circumference of a circle is difficult. And all because she has no corners. With them, everything would be much easier.

What is a circle and where does it occur?

This flat figure is 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, except for people who are engineers and designers. They design 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 is absolutely impossible to calculate the circumference of a circle with absolute accuracy. This is due to the infinity of the main number in the formula. "Pi" is still being specified. And most often the rounded value is used. The degree of accuracy is chosen so as to give the most correct answer.

Notation of quantities and formulas

Now it is easy to answer the question of how to calculate the circumference of a circle from a radius, this will require the following formula:

Since the 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 \u003d π * d.

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

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

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

Task examples

Task one

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

Solution. Here it is easy to understand how to calculate the circumference of a circle. You just need to use the first formula. Since the radius is known, all you need to do is plug in the values ​​and count. 2 multiplied by a radius of 5 cm gives 10. It remains 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. You need 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 remains: 200. This is the desired value.

Answer: r = 200 mm.

Task three

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

Solution. Similar to the previous problem, you need to divide the known length by the value of π, rounded up to hundredths. As a result of such an action, the number 18 is obtained. The result is obtained.

Answer: d = 18 cm.

Task four

Condition. The clock hands are 3 and 5 cm long. It is necessary 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. It needs to be used twice.

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

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

Solution. This distance is equal to the circumference of the circle. Therefore, you need to use the appropriate formula. Namely, multiply the value of π and 2 m. The calculations give the result: 6.28 m.

Answer: Squirrel runs 6.28 m.

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.

A circle is a curved line that encloses a circle. In geometry, figures are flat, so the definition refers to a two-dimensional image. It is assumed that all points of this curve are at an equal distance from the center of the circle.

The circle has several characteristics, on the basis of which the calculations associated with this geometric figure are made. These include: diameter, radius, area and circumference. These characteristics are interrelated, that is, information about at least one of the components is sufficient to calculate them. For example, knowing only the radius of a geometric figure using the formula, you can find the circumference, diameter, and its area.

• The radius of a circle is a segment inside the circle connected to its center.
• Diameter is a line segment inside a circle that connects its points and passes through the center. In fact, the diameter is two radii. This is exactly what the formula for calculating it looks like: D=2r.
• There is another component of the circle - the chord. This is a straight line that connects two points on a circle, but does not always pass through the center. So the chord that passes through it is also called the diameter.

How to find the circumference of a circle? Now let's find out.

Circumference: formula

The Latin letter p has been chosen to designate this characteristic. Archimedes also proved that the ratio of the circumference of a circle to its diameter is the same number for all circles: it is the number π, which is approximately equal to 3.14159. The formula for calculating π looks like this: π = p/d. According to this formula, the value of p is equal to πd, that is, the circumference: p= πd. Since d (diameter) is equal to two radii, the same circumference formula can be written as p=2πr. Consider the application of the formula using simple problems as an example:

Task 1

At the base of the Tsar Bell, the diameter is 6.6 meters. What is the circumference of the base of the bell?

1. So, the formula for calculating the circle is p= πd
2. We substitute the existing value in the formula: p \u003d 3.14 * 6.6 \u003d 20.724

Answer: The circumference of the base of the bell is 20.7 meters.

Task 2

An artificial satellite of the Earth rotates at a distance of 320 km from the planet. The radius of the Earth is 6370 km. What is the length of the satellite's circular orbit?

1. 1. Calculate the radius of the circular orbit of the Earth satellite: 6370+320=6690 (km)
2. 2. Calculate the length of the circular orbit of the satellite using the formula: P=2πr
3. 3.P=2*3.14*6690=42013.2

Answer: the length of the circular orbit of the Earth's satellite is 42013.2 km.

Methods for measuring the circumference

The calculation of the circumference of a circle is not often used in practice. The reason for this is the approximate value of the number π. In everyday life, a special device is used to find the length of a circle - a curvimeter. An arbitrary reference point is marked on the circle and the device is guided from it strictly along the line until they again reach this point.

How to find the circumference of a circle? You just need to keep in mind simple formulas for calculations.

Instruction

Recall that Archimedes first calculated this ratio mathematically. It is regular 96-gons inside and around the circle. The perimeter of the inscribed polygon was taken as the minimum possible circumference, the perimeter of the circumscribed figure was taken as the maximum size. According to Archimedes, the ratio of circumference to diameter is 3.1419. Much later, this number was "lengthened" to eight digits by the Chinese mathematician Zu Chongzhi. His calculations remained the most accurate for 900 years. In the 18th century alone, one hundred decimal places were counted. And since 1706, this infinite decimal fraction, thanks to William Jones, has acquired a name. He designated it with the first letter of the Greek words perimeter (periphery). Today, the computer easily calculates the signs of the number Pi: ​​3.141592653589793238462643 ...

For calculations, reduce Pi to 3.14. It turns out that for any circle its length divided by the diameter is equal to this number: L:d=3.14.

Express from this statement a formula for finding the diameter. It turns out that to find the diameter of a circle, you need to divide the circumference by pi. It looks like this: d = L:3.14. This is a universal way to find the diameter when the circumference of a circle is known.

So, the circumference is known, let's say 15.7 cm, divide this figure by 3.14. The diameter will be 5 cm. Write it like this: d \u003d 15.7: 3.14 \u003d 5 cm.

Find the diameter from the circumference using special tables for calculating the circumference. These tables are included in various reference books. For example, they are in the "Four-digit mathematical tables" by V.M. Bradis.

Useful advice

Memorize the first eight digits of pi with a poem:
You just have to try
And remember everything as it is:
Three, fourteen, fifteen
Ninety-two and six.

Sources:

• The number "Pi" is calculated with record accuracy
• diameter and circumference
• How to find the circumference of a circle?

A circle is a flat geometric figure, all points of which are at the same and non-zero distance from the selected point, which is called the center of the circle. A straight line connecting any two points of a circle and passing through the center is called it. diameter. The total length of all the boundaries of a two-dimensional figure, which is usually called the perimeter, for a circle is more often denoted as the "circumference". Knowing the circumference of a circle, you can calculate its diameter.

Instruction

Use one of the basic properties of a circle to find the diameter, which is that the ratio of the length of its perimeter to the diameter is the same for absolutely all circles. Of course, constancy did not go unnoticed by mathematicians, and this proportion has long since received its own - this is the number Pi (π is the first Greek word " circle" and "perimeter"). The numerical value of this is determined by the circumference of a circle whose diameter is equal to one.

Divide the known circumference of a circle by pi to calculate its diameter. Since this number is "", it does not have a finite value - it is a fraction. Round pi according to the accuracy of the result you need to get.

Related videos

Tip 4: How to find the ratio of the circumference of a circle to the length of the diameter

Amazing Property circles opened to us by the ancient Greek scientist Archimedes. It lies in the fact that attitude her length to the length of the diameter is the same for any circles. In his work "On the measurement of the circle" he calculated it and designated it as the number "Pi". It is irrational, that is, its meaning cannot be precisely expressed. For, its value equal to 3.14 is used. You can verify Archimedes' statement yourself by doing simple calculations.

You will need

• - compass;
• - ruler;
• - pencil;
• - thread.

Instruction

Draw a circle of arbitrary diameter on paper with a compass. Using a ruler and a pencil, draw a segment through its center connecting the two located on the line circles. Use a ruler to measure the length of the resulting segment. Let's say circles in this case, 7 centimeters.

Take the thread and arrange it along the length circles. Measure the resulting thread length. Let it be equal to 22 centimeters. Find attitude length circles to the length of its diameter - 22 cm: 7 cm \u003d 3.1428 .... Round the resulting number (3.14). It turned out the familiar number "Pi".

Prove this property circles you can, using a cup or glass. Measure their diameter with a ruler. Wrap the top of the dish with a thread, measure the resulting length. Dividing the length circles cup by the length of its diameter, you will also get the number "Pi", making sure of this property circles discovered by Archimedes.

Using this property, you can calculate the length of any circles along the length of its diameter or according to the formulas: C \u003d 2 * p * R or C \u003d D * p, where C - circles, D - the length of its diameter, R - the length of its radius. To find (the plane bounded by lines circles) use the formula S = π*R² if its radius is known, or the formula S = π*D²/4 if its diameter is known.

note

Did you know that March 14 has been Pi Day for more than twenty years? This is an unofficial holiday of mathematicians dedicated to this interesting number, with which many formulas, mathematical and physical axioms are currently associated. This holiday was invented by the American Larry Shaw, who noticed that on this day (3.14 in the US date system) the famous scientist Einstein was born.

Sources:

• Archimedes

Sometimes a convex polygon can be drawn in such a way that the vertices of all corners lie on it. Such a circle with respect to the polygon should be called circumscribed. Her center does not have to be inside the perimeter of the inscribed figure, but using the properties of the described circles, finding this point is usually not very difficult.

You will need

• Ruler, pencil, protractor or square, compasses.

Instruction

If the polygon around which you want to describe the circle is drawn on paper, to find center and a circle is enough for a ruler, pencil and protractor or square. Measure the length of any of the sides of the figure, determine its middle and put an auxiliary point in this place of the drawing. Using a square or a protractor, draw a segment perpendicular to this side inside the polygon until it intersects with the opposite side.

Do the same operation with any other side of the polygon. The intersection of the two constructed segments will be the desired point. This follows from the main property of the described circles- her center in a convex polygon with any side always lies at the point of intersection of the perpendicular bisectors drawn to these .

For regular polygons center but inscribed circles could be much easier. For example, if it is a square, then draw two diagonals - their intersection will be center ohm inscribed circles. In a polygon with any even number of sides, it is enough to connect two pairs of opposite corners with auxiliary ones - center described circles must coincide with the point of their intersection. In a right triangle, to solve the problem, simply determine the middle of the longest side of the figure - the hypotenuse.

If it is not known from the conditions whether, in principle, the circumscribed circle for a given polygon is possible, after determining the supposed point center and by any of the methods described, you can find out. Set aside on the compass the distance between the found point and any of , set to the estimated center circles and draw a circle - each vertex must lie on this circles. If this is not the case, then one of the properties is not satisfied and describe a circle around the given polygon.

Determining the diameter can be useful not only for solving geometric problems, but also to help in practice. For example, knowing the diameter of the neck of a jar, you will definitely not make a mistake in choosing a lid for it. The same statement is true for larger circles.

Instruction

So, enter the notation for the quantities. Let d be the diameter of the well, L be the circumference, n be the Pi number, which is approximately equal to 3.14, R be the radius of the circle. The circumference (L) is known. Let's assume that it is equal to 628 centimeters.

Next, to find the diameter (d), use the formula for the circumference: L=2nR, where R is an unknown value, L=628 cm, and n=3.14. Now use the rule for finding an unknown factor: "To find a factor, you need to divide the product by a known factor." It turns out: R \u003d L / 2p. Substitute the values ​​into the formula: R=628/2x3.14. It turns out: R=628/6.28, R=100 cm.

After the radius of the circle is found (R=100 cm), use the following formula: the diameter of the circle (d) is equal to two radii of the circle (2R). It turns out: d=2R.

Now, to find the diameter, substitute the values ​​​​in the formula d \u003d 2R and calculate the result. Since the radius (R) is known, it turns out: d=2x100, d=200 cm.

Sources:

• how to find the diameter of a circle

The circumference and diameter are interrelated geometric quantities. This means that the first of them can be translated into the second one without any additional data. The mathematical constant through which they are interconnected is the number π.

Instruction

If the circle is represented as an image on paper, and you want to determine its diameter approximately, measure it directly. If its center is shown in the drawing, draw a line through it. If the center is not shown, find it with a compass. To do this, use a square with angles of 90 and. Attach it with a 90-degree angle to the circle so that both legs touch it, and circle. Attaching then to the resulting right angle a 45-degree angle of the square, draw. It will pass through the center of the circle. Then, in a similar way, draw a second right angle and its bisector in another place on the circle. They intersect in the center. This will measure the diameter.

To measure the diameter, it is preferable to use a ruler made of the thinnest sheet material possible, or a tailor's meter. If you have only a thick ruler, measure the diameter of the circle with a compass, and then, without changing its solution, transfer it to graph paper.

Also, in the absence of numerical data in the conditions of the problem and with only a drawing, you can measure the circumference using a curvimeter, and then calculate the diameter. To use the curvimeter, first rotate its wheel to set the pointer exactly to zero division. Then mark a point on the circle and press the meter against the sheet so that the stroke above the wheel points to this point. Move the wheel along the circle line until the stroke is again over this point. Read the statements. They will be in bounded by a broken line. If a regular n-gon with side b is inscribed in a circle, then the perimeter of such a figure P is equal to the product of side b by the number of sides n: P \u003d b * n. Side b can be determined by the formula: b=2R*Sin (π/n), where R is the radius of the circle into which the n-gon is inscribed.

As the number of sides increases, the perimeter of the inscribed polygon will increasingly approach L. Р= b*n=2n*R*Sin (π/n)=n*D*Sin (π/n). The relationship between the circumference L and its diameter D is constant. The ratio L / D \u003d n * Sin (π / n) as the number of sides of the inscribed polygon tends to infinity tends to the number π, a constant value called "pi" and expressed as an infinite decimal fraction. For calculations without the use of computer technology, the value π=3.14 is taken. The circumference of a circle and its diameter are related by the formula: L= πD. To calculate the diameter

Circumference measurement

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 of the 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 FROM, 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.

Because 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 ( FROM) 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 FROM, 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.

A 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 FROM = 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 the full surface of the cylinder should be calculated. 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.