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.

Helpful 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 test 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 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 intersection point mid-perpendiculars held 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 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. Applying then to the resulting right angle 45-degree angle of a 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 number" and pronounced infinite decimal. For calculations without using computer science 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

The fact that our planet has the shape of a ball has been known to scientists engaged in research in the field of geology for a long time. That is why the first measurements of the circumference earth's surface touched the longest parallel of the Earth - the equator. This value, scientists believed, can be considered correct for any other method of measurement. For example, it was believed that if you measure the circumference of the planet by the longest meridian, the resulting figure will be exactly the same.

This view continued until the 18th century. However, scientists from the leading scientific institution of that time - the French Academy - were of the opinion that this hypothesis is incorrect, and the shape that the planet has is not entirely correct. Therefore, in their opinion, the circumferences along the longest meridian and along the longest parallel will differ.

As proof, two scientific expeditions were undertaken in 1735 and 1736, which proved the truth of this assumption. Subsequently, the magnitude of the difference between these two was also established - it amounted to 21.4 kilometers.

Circumference

At present, the circumference of the planet Earth has been repeatedly measured not by extrapolating the length of one or another segment of the earth's surface to its full size, as was done before, but by using modern high-precision technologies. Thanks to this, it was possible to establish the exact circumference along the longest meridian and the longest parallel, as well as to clarify the magnitude of the difference between these parameters.

So, today in the scientific community, as the official value of the circumference of the planet Earth along the equator, that is, the longest parallel, it is customary to give a figure of 40075.70 kilometers. At the same time, a similar parameter measured along the longest meridian, that is, the circumference passing through the earth's poles, is 40,008.55 kilometers.

Thus, the difference between the circumferences is 67.15 kilometers, and the equator is the longest circle on our planet. In addition, the difference means that one degree of the geographic meridian is somewhat shorter than one degree of the geographic parallel.

Instruction

At first it is necessary the initial data to the task. The fact is that its condition cannot be explicitly said what is the radius circles. Instead, the problem can be given the length of the diameter circles. Diameter circles a line segment that connects two opposite points circles passing through its center. Having analyzed the definitions circles, we can say that the length of the diameter is twice the length of the radius.

Now we can accept the radius circles equal to R. Then for the length circles you need to use the formula:
L = 2πR = πD, where L is the length circles, D - diameter circles, which is always 2 times the radius.

note

A circle can be inscribed in a polygon, or described around it. Moreover, if the circle is inscribed, then it will divide them in half at the points of contact with the sides of the polygon. To find the radius of an inscribed circle, you need to divide the area of ​​the polygon by half its perimeter:
R = S/p.
If a circle is circumscribed around a triangle, then its radius is found by the following formula:
R \u003d a * b * c / 4S, where a, b, c are the sides of the given triangle, S is the area of ​​\u200b\u200bthe triangle around which the circle is described.
If it is required to describe a circle around a quadrilateral, then this can be done subject to two conditions:
The quadrilateral must be convex.
The sum of the opposite angles of the quadrilateral should be 180°

Helpful advice

In addition to the traditional caliper, stencils can also be used to draw a circle. In modern stencils, a circle of different diameters is included. These stencils can be purchased at any stationery store.

Sources:

• How to find the circumference of a circle?

Circle - a closed curved line, all points of which are at an equal distance from one point. This point is the center of the circle, and the segment between the point on the curve and its center is called the radius of the circle.

Instruction

If a straight line is drawn through the center of a circle, then its segment between the two points of intersection of this line with the circle is called the diameter of this circle. Half the diameter, from the center to the point where the diameter intersects with the circle, is the radius
circles. If the circle is cut at an arbitrary point, straightened and measured, then the resulting value is the length of the given circle.

Draw several circles with different compass solutions. Visual comparison leads to the conclusion that a larger diameter outlines a larger circle bounded by a circle with a larger length. Therefore, there is a directly proportional relationship between the diameter of a circle and its length.

By physical meaning the parameter "circumference" corresponds to , limited 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 number” 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. For a circle, divide its length by π=3.14.

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. Let's 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.

A circle is a series of points equidistant from one point, which, in turn, is the center of this circle. The circle also has its own radius, equal to the distance of these points from the center.

The ratio of the length of a circle to its diameter is the same for all circles. This ratio is a number that is a mathematical constant, which is denoted by the Greek letter π .

Determining the circumference of a circle

You can calculate the circle using the following formula:

L= π D=2 π r

r- circle radius

D- circle diameter

L- circumference

π - 3.14

Task:

Calculate circumference with a radius of 10 centimeters.

Solution:

Formula for calculating the dyne of a circle looks like:

L= π D=2 π r

where L is the circumference, π is 3.14, r is the radius of the circle, D is the diameter of the circle.

Thus, the circumference of a circle with a radius of 10 centimeters is:

L = 2 × 3.14 × 10 = 62.8 centimeters

Circle is a geometric figure, which is a collection of all points on the plane, remote from a given point, which is called its center, at a distance that is not equal to zero and is called the radius. Scientists knew how to determine its length with varying degrees of accuracy already in ancient times: historians of science believe that the first formula for calculating the circumference of a circle was compiled around 1900 BC in ancient Babylon.

With such geometric shapes like circles we collide daily and everywhere. It is its shape that has the outer surface of the wheels, which are equipped with various vehicles. This detail, despite its outward simplicity and unpretentiousness, is considered one of the greatest inventions of mankind, and it is interesting that the natives of Australia and American Indians until the arrival of the Europeans, they had absolutely no idea what it was.

In all likelihood, the very first wheels were pieces of logs that were mounted on an axle. Gradually, the design of the wheel improved, their design became more and more complex, and for their manufacture it was necessary to use a lot of different tools. First, wheels appeared, consisting of a wooden rim and spokes, and then, in order to reduce wear on their outer surface, they began to upholster it with metal strips. In order to determine the lengths of these elements, it is necessary to use the formula for calculating the circumference (although in practice, most likely, the craftsmen did this “by eye” or simply girding the wheel with a strip and cutting off the required section of it).

It should be noted that wheel is used not only in vehicles. For example, a potter's wheel has its shape, as well as elements of gears of gears widely used in technology. Since ancient times, wheels have been used in the construction of water mills (the oldest structures of this kind known to scientists were built in Mesopotamia), as well as spinning wheels used to make threads from animal wool and plant fibers.

circles often found in construction. Their shape is quite widespread round windows, very characteristic of the Romanesque architectural style. The manufacture of these structures is a very difficult task and requires high skill, as well as the availability of a special tool. One of the varieties of round windows are portholes installed in ships and aircraft.

Thus, design engineers often have to solve the problem of determining the circumference of a circle, developing various machines, mechanisms and assemblies, as well as architects and designers. Since the number π necessary for this is infinite, then it is not possible to determine this parameter with absolute accuracy, and therefore, the calculations take into account that degree of it, which in a particular case is necessary and sufficient.

If the problem knows such quantities as the circumference of a circle, its radius or the area of ​​a circle that is bounded by a given circle, then the calculation of the diameter will be simple. There are several ways to calculate the diameter of a circle. They are quite simple and do not cause any difficulties at all, as it seems to many at first glance.

How to find the diameter of a circle - 1 way

When the value of the radius of the circle is given, then the problem can be considered half solved, since the radius is the distance from a point that lies anywhere on the circle to the center of this very circle. All that needs to be done to find the diameter in this case is to multiply the given radius value by 2. This way of calculating is because the radius is half the diameter. Therefore, if it is known what the radius is equal to, then the value of half of the desired diameter value has actually been found.

How to find the diameter of a circle - 2 way

If the problem is given only the value of the circumference of a circle, then to find the value of the diameter, you just need to divide it by a number known as π, the approximate value of which is 3.14. That is, if the length value is 31.4, then dividing it by 3.14, we get the diameter value, which is 10.

How to find the diameter of a circle - 3 way

If the value of the area of ​​the circle is given in the source data, then it is also easy to find the diameter. All you have to do is extract Square root from this value and divide the result by the number π. This means that if the area value is 64, then when extracting the root, the number 8 remains. If we divide the resulting 8 by 3.14, we get the diameter value, which is approximately 2.5.

How to find the diameter of a circle - 4 way

Inside the circle, you need to draw a straight horizontal line from one point to another using a ruler or square. Mark the intersections of this line with the circle line with letters, for example, A and B. It does not matter in which part of the circle this line will be located.

After that, you need to draw two more circles. But in such a way that points A and B become their centers. Again educated figures will intersect at two points. Through them you need to draw another straight line. After that, we measure its length with a ruler. The measurement value will be equal to the length of the diameter, because the last line drawn is the diameter itself.

It is interesting that even very far in the past, for weaving baskets of a certain size, twigs were taken about 3 times longer. Scientists have explained and proved experimentally that if the length of any circle is divided by the diameter, then the result is almost the same number.