A circle is a closed curve, all points of which are at the same distance from the center. This figure is flat. Therefore, the solution to the problem, the question of which is how to find the circumference of a circle, is quite simple. All available methods, we will consider in today's article.
Figure descriptions
In addition to a fairly simple descriptive definition, there are three more mathematical characteristics of a circle, which in themselves contain the answer to the question of how to find the circumference of a circle:
- Consists of points A and B and all others from which AB can be seen at right angles. Diameter of this figure equal to length the section under consideration.
- Includes only points X such that the ratio AX/BX is constant and not equal to one. If this condition is not met, then it is not a circle.
- It consists of points, for each of which the following equality holds: the sum of the squared distances to the other two is a given value, which is always greater than half the length of the segment between them.
Terminology
Not everyone at school had a good math teacher. Therefore, the answer to the question of how to find the circumference of a circle is also complicated by the fact that not everyone knows the basic geometric concepts. Radius - a segment that connects the center of the figure with a point on the curve. A special case in trigonometry is the unit circle. A chord is a line segment that connects two points on a curve. For example, the already considered AB falls under this definition. Diameter is a chord passing through the center. The number π is equal to the length of the unit semicircle.
Basic formulas
Geometric formulas directly follow from the definitions, which allow you to calculate the main characteristics of the circle:
- The length is equal to the product of the number π and the diameter. The formula is usually written as follows: C = π*D.
- The radius is half the diameter. It can also be calculated by calculating the quotient of dividing the circumference by twice the number π. The formula looks like this: R = C/(2* π) = D/2.
- The diameter is equal to the circumference divided by π or twice the radius. The formula is quite simple and looks like this: D = C/π = 2*R.
- The area of a circle is equal to the product of the number π and the square of the radius. Similarly, diameter can be used in this formula. In this case, the area will be equal to the quotient of dividing the product of the number π and the square of the diameter by four. The formula can be written as follows: S = π*R 2 = π*D 2 /4.
How to find the circumference of a circle from a diameter
For simplicity of explanation, we denote by letters the characteristics of the figure necessary for calculating. Let C be the desired length, D be its diameter, and let pi be approximately 3.14. If we have only one known quantity, then the problem can be considered solved. Why is it necessary in life? Suppose we decide to enclose a round pool with a fence. How to calculate required amount columns? And here the ability to calculate the circumference of a circle comes to the rescue. The formula is as follows: C = π D. In our example, the diameter is determined based on the radius of the pool and the required distance to the fence. For example, suppose that our home artificial pond is 20 meters wide, and we are going to put posts at a distance of ten meters from it. The diameter of the resulting circle is 20 + 10 * 2 = 40 m. The length is 3.14 * 40 = 125.6 meters. We will need 25 columns if the gap between them is about 5 m.
Length through radius
As always, let's start by assigning letter circles to characteristics. In fact, they are universal, so mathematicians from different countries it is not necessary to know each other's language. Suppose C is the circumference of a circle, r is its radius, and π is approximately 3.14. The formula looks like this in this case: C = 2*π*r. Obviously, this is an absolutely correct equality. As we have already figured out, the diameter of a circle is equal to twice its radius, so this formula looks like this. In life, this method can also often come in handy. For example, we bake a cake in a special sliding form. So that it does not get dirty, we need a decorative wrapper. But how to cut a circle right size. This is where mathematics comes to the rescue. Those who know how to find out the circumference of a circle will immediately say that you need to multiply the number π by twice the radius of the shape. If its radius is 25 cm, then the length will be 157 centimeters.
Task examples
We have already considered several practical cases of the acquired knowledge on how to find out the circumference of a circle. But often we are not concerned with them, but with real ones. math problems contained in the textbook. After all, the teacher gives points for them! Therefore, let's consider a problem of increased complexity. Let's assume that the circumference is 26 cm. How to find the radius of such a figure?
Example Solution
To begin with, let's write down what is given to us: C \u003d 26 cm, π \u003d 3.14. Also remember the formula: C = 2* π*R. From it you can extract the radius of the circle. Thus, R= C/2/π. Now let's proceed to the direct calculation. First, divide the length by two. We get 13. Now we need to divide by the value of the number π: 13 / 3.14 \u003d 4.14 cm. It is important not to forget to write down the answer correctly, that is, with units of measurement, otherwise the whole practical meaning of such problems is lost. In addition, for such inattention, you can get a score of one point lower. And no matter how annoying it may be, you have to put up with this state of affairs.
The beast is not as scary as it is painted
So we figured out such a difficult task at first glance. As it turned out, you just need to understand the meaning of the terms and remember a few easy formulas. Math is not so scary, you just need to make a little effort. So geometry is waiting for you!
A circle is made up of many points that are equidistant from the center. This is a flat geometric figure, and finding its length is not difficult. A person encounters a circle and a circle every day, regardless of the area in which he works. Many vegetables and fruits, devices and mechanisms, utensils and furniture have round shape. A circle is a set of points that is within the boundaries of a circle. Therefore, the length of the figure is equal to the perimeter of the circle.
Characteristics of the figure
In addition to the fact that the description of the concept of a circle is quite simple, its characteristics are also easy to understand. With their help, you can calculate its length. The inner part of the circle consists of many points, among which two - A and B - can be seen at right angles. This segment is called the diameter, it consists of two radii.
Within the circle there are points X such, which does not change and does not equal unity, the ratio AX / BX. In a circle, this condition is necessarily observed, otherwise this figure does not have the shape of a circle. The rule applies to each point that makes up the figure: the sum of the squared distances from these points to two others always exceeds half the length of the segment between them.
Basic circle terms
In order to be able to find the length of a figure, you need to know the basic terms related to it. The main parameters of the figure are diameter, radius and chord. A radius is a segment that connects the center of a circle with any point on its curve. The value of a chord is equal to the distance between two points on the curved figure. Diameter - distance between points passing through the center of the figure.
Basic formulas for calculations
The parameters are used in the formulas for calculating the values of the circle:
Diameter in calculation formulas
In economics and mathematics, it often becomes necessary to find the circumference of a circle. But also in Everyday life you may encounter this need, for example, during the construction of a fence around a round pool. How to calculate the circumference of a circle from a diameter? In this case, use the formula C \u003d π * D, where C is the desired value, D is the diameter.
For example, the width of the pool is 30 meters, and the fence posts are planned to be placed at a distance of ten meters from it. In this case, the formula for calculating the diameter is: 30+10*2 = 50 meters. The desired value (in this example, the length of the fence): 3.14 * 50 \u003d 157 meters. If the fence posts stand at a distance three meters from each other, then a total of 52 will be needed.
Radius calculations
How to calculate the circumference of a circle from a known radius? For this, the formula C \u003d 2 * π * r is used, where C is the length, r is the radius. The radius in a circle is less than half the diameter, and this rule can come in handy in everyday life. For example, in the case of making a pie in a sliding form.
In order for the culinary product not to get dirty, it is necessary to use a decorative wrapper. And how to cut a paper circle of a suitable size?
Those who are a little familiar with mathematics understand that in this case you need to multiply the number π by twice the radius of the shape used. For example, the diameter of the mold is 20 centimeters, respectively, its radius is 10 centimeters. These parameters are required size circles: 2 * 10 * 3, 14 \u003d 62.8 centimeters.
Handy calculation methods
If it is not possible to find the circumference using the formula, then you should use the available methods for calculating this value:
- With a small round object, its length can be found using a rope wrapped around once.
- The size of a large object is measured as follows: a rope is laid out on a flat plane, and a circle is rolled over it once.
- Modern students and schoolchildren use calculators for calculations. Known parameters can be used to find out unknown values online.
Round objects in the history of human life
The first round product that man invented was the wheel. The first structures were small rounded logs mounted on axles. Then came wheels made of wooden spokes and rims. Gradually added to the product metal parts to reduce wear. It was in order to find out the length of the metal strips for the upholstery of the wheel that scientists of past centuries were looking for a formula for calculating this value.
It has the shape of a wheel Potter's wheel , most of the details in complex mechanisms, designs of water mills and spinning wheels. Often there are round objects in construction - the frames of round windows in the Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers daily in the field of their professional activities are faced with the need to calculate the size of a circle.
Many objects in the world around us are round. These are wheels, round window openings, pipes, various utensils and much more. You can calculate the circumference of a circle by knowing its diameter or radius.
There are several definitions of this geometric figure.
- It 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 does not equal 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 as well. A circle is an area within a circle. The perimeter of a circle is its length. According to various 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 with a simple formula:
where L is the desired value,
π is the number pi, approximately equal to 3.1413926.
Usually, to find the desired value, it is enough to use π up to the second decimal place, that is, 3.14, this will provide the desired accuracy. On calculators, in particular engineering ones, there may be a button that automatically enters the value of the number π.
Notation
To find through the diameter, there is the following formula:
If L is already known, you can easily find out the radius or diameter. To do this, L must be divided by 2π or π, respectively.
If a circle is already given, you need to understand how to find the circumference from this data. The area of a 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)
Summarizing, we can say that there are three main formulas:
- through the radius – L = 2πR;
- through the diameter - L = πD;
- through the area of a circle – L = 2√(Sπ).
Pi
Without the number π, it will not be possible to solve the problem under consideration. The number π was found for the first time 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 now 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 considered not only from the point of view of geometry, but also from the point of view of mathematical analysis through the sums of series. The notation for this constant with the Greek letter π was first used by William Jones in 1706, and became popular after the work of Euler.
It is now known that this constant is an infinite non-periodic decimal fraction, it is irrational, that is, it cannot be represented as a ratio of two integers. With the help of calculations on supercomputers in 2011, they learned the 10-trillion sign of a constant.
It is interesting! To memorize the first few characters of the number π, various mnemonic rules were invented. Some allow you to store a large number of digits in memory, for example, one French poem will help you remember pi up to 126 characters.
If you need the circumference, the online calculator will help you with this. There are many such calculators, they only 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 accuracy, you need to specify the number of decimal places. Also, using online calculators, you can calculate the area of a circle.
Such calculators are easy to find with any search engine. There are also mobile applications that 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 come in handy. For example, it is required to wrap a cake baked in a form with a diameter of 20 cm with a paper strip. Then it will not be difficult to find the length of this strip:
L \u003d πD \u003d 3.14 * 20 \u003d 62.8 cm.
Another example: you need to build a fence around a circular 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 \u003d 2πR \u003d 2 * 3.14 * 13 \u003d 81.68 m.
Useful video: circle - radius, diameter, circumference
Outcome
The perimeter of a circle can be easily calculated from simple formulas, including diameter or radius. You can also find the desired value through the area of the circle. Online calculators or mobile applications will help to solve this problem, in which you need to enter a single number - diameter or radius.
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 some distance, not equal to zero and 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 figures as circles, we encounter 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 the American Indians, until the arrival of the Europeans, 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 mass various tools. First, wheels appeared, consisting of a wooden rim and spokes, and then, in order to reduce wear on them 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 used by no means 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.
