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, is quite simple. We will look at all available methods 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:
- Consists of points A and B and all others from which AB can be seen at right angles. The diameter of this figure is equal to the length of the segment under consideration.
- Includes only those 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 squares of the distances to the other two is a given value, which is always more 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 further complicated by the fact that not everyone knows the basic geometric concepts. Radius is a segment that connects the center of a figure to a point on a curve. A special case in trigonometry is unit circle. A chord is a segment that connects two points on a curve. For example, the already discussed AB falls under this definition. The diameter is the chord passing through the center. The number π is equal to the length of a unit semicircle.
Basic formulas
From the definitions it follows directly geometric formulas, which allow you to calculate the main characteristics of a 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 equal to 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 quotient of 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 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 the product of π and the square of the diameter divided 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 by diameter
To simplify the explanation, let us denote by letters the characteristics of the figure necessary for the calculation. Let C be the desired length, D its diameter, and π approximately equal to 3.14. If we have only one known quantity, then the problem can be considered solved. Why is this necessary in life? Suppose we decide to surround a round pool with a fence. How to calculate required quantity columns? And here the ability to calculate the circumference 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 from the fence. For example, suppose that our home artificial pond is 20 meters wide, and we are going to place the posts at a ten-meter distance from it. The diameter of the resulting circle is 20 + 10*2 = 40 m. Length is 3.14*40 = 125.6 meters. We will need 25 posts if the gap between them is about 5 m.
Length through radius
As always, let's start by assigning letters to the characteristics of the circle. In fact, they are universal, so mathematicians from different countries It is not at all necessary to know each other's language. Let's assume that C is the circumference of the circle, r is its radius, and π is approximately equal to 3.14. The formula in this case looks like this: C = 2*π*r. Obviously, this is an absolutely correct equation. 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. To prevent it from getting dirty, we need a decorative wrapper. But how to cut a circle of the required 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.
Sample problems
We have already looked at several practical cases of acquired knowledge on how to find out the circumference of a circle. But often we are not concerned about them, but about real math problems which are contained in the textbook. After all, the teacher gives points for them! So let's look at a more complex problem. Let's assume that the circumference of the circle is 26 cm. How to find the radius of such a figure?
Example solution
First, let's write down what we are given: C = 26 cm, π = 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 actual 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 = 4.14 cm. It is important not to forget to write the answer correctly, that is, with units of measurement, otherwise the entire practical meaning of such problems is lost. In addition, for such inattention you can get a grade one point lower. And no matter how annoying it may be, you will have to put up with this state of affairs.
The beast is not as scary as it is painted
So we have dealt with such a difficult task at first glance. As it turns out, you just need to understand the meaning of the terms and remember a few simple formulas. Math is not that scary, you just need to put in a little effort. So geometry is waiting for you!
It often sounds like part of a plane that is bounded by a circle. The circumference of a circle is a flat closed curve. All points located on the curve are the same distance from the center of the circle. In a circle, its length and perimeter are the same. The ratio of the length of any circle and its diameter is constant and is denoted by the number π = 3.1415.
Determining the perimeter of a circle
The perimeter of a circle of radius r is equal to twice the product of radius r and the number π(~3.1415)
Circle perimeter formula
Perimeter of a circle of radius \(r\) :
\[ \LARGE(P) = 2 \cdot \pi \cdot r \]
\[ \LARGE(P) = \pi \cdot d \]
\(P\) – perimeter (circumference).
\(r\) – radius.
\(d\) – diameter.
We will call a circle a geometric figure that consists of all such points that are at the same distance from any given point.
Center of the circle we will call the point that is specified within Definition 1.
Circle radius we will call the distance from the center of this circle to any of its points.
In the Cartesian coordinate system \(xOy\) we can also introduce the equation of any circle. Let us denote the center of the circle by the point \(X\) , which will have coordinates \((x_0,y_0)\) . Let the radius of this circle be equal to \(τ\) . Let's take an arbitrary point \(Y\) whose coordinates we denote by \((x,y)\) (Fig. 2).
Using the formula for the distance between two points in our given coordinate system, we get:
\(|XY|=\sqrt((x-x_0)^2+(y-y_0)^2) \)
On the other hand, \(|XY| \) is the distance from any point on the circle to the center we have chosen. That is, by definition 3, we obtain that \(|XY|=τ\) , therefore
\(\sqrt((x-x_0)^2+(y-y_0)^2)=τ \)
\((x-x_0)^2+(y-y_0)^2=τ^2 \) (1)
Thus, we get that equation (1) is the equation of a circle in the Cartesian coordinate system.
Circumference (perimeter of a circle)
We will derive the length of an arbitrary circle \(C\) using its radius equal to \(τ\) .
We will consider two arbitrary circles. Let us denote their lengths by \(C\) and \(C"\) , whose radii are equal to \(τ\) and \(τ"\) . We will inscribe regular \(n\)-gons into these circles, the perimeters of which are equal to \(ρ\) and \(ρ"\), the lengths of the sides are equal to \(α\) and \(α"\), respectively. As we know, the side of a regular \(n\) square inscribed in a circle is equal to
\(α=2τsin\frac(180^0)(n) \)
Then we get that
\(ρ=nα=2nτ\frac(sin180^0)(n) \)
\(ρ"=nα"=2nτ"\frac(sin180^0)(n) \)
\(\frac(ρ)(ρ")=\frac(2nτsin\frac(180^0)(n))(2nτ"\frac(sin180^0)(n))=\frac(2τ)(2τ" ) \)
We get that the relationship \(\frac(ρ)(ρ")=\frac(2τ)(2τ") \) will be true regardless of the number of sides of the inscribed regular polygons. That is
\(\lim_(n\to\infty)(\frac(ρ)(ρ"))=\frac(2τ)(2τ") \)
On the other hand, if we infinitely increase the number of sides of inscribed regular polygons (that is, \(n→∞\)), we obtain the equality:
\(lim_(n\to\infty)(\frac(ρ)(ρ"))=\frac(C)(C") \)
From the last two equalities we obtain that
\(\frac(C)(C")=\frac(2τ)(2τ") \)
\(\frac(C)(2τ)=\frac(C")(2τ") \)
We see that the ratio of the circumference of a circle to its double radius is always the same number, regardless of the choice of the circle and its parameters, that is
\(\frac(C)(2τ)=const \)
This constant should be called the number “pi” and denoted \(π\) . Approximately, this number will be equal to \(3.14\) ( exact value this number does not exist, since it is an irrational number). Thus
\(\frac(C)(2τ)=π \)
Finally, we find that the circumference (perimeter of a circle) is determined by the formula
\(C=2πτ\)
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A ruler alone is not enough; you need to know special formulas. The only thing we need to do is determine the diameter or radius of the circle. In some problems these quantities are indicated. But what if we have nothing but a drawing? No problem. The diameter and radius can be calculated using a regular ruler. Now let's get down to the basics.
Formulas everyone should know
Almost 4,000 years ago, scientists discovered an amazing relationship: if the circumference of a circle is divided by its diameter, the result is the same number, which is approximately 3.14. This meaning was named with this letter in the ancient Greek language, the words “perimeter” and “circumference” began. Based on the discovery made by ancient scientists, you can calculate the length of any circle:
Where P means the length (perimeter) of the circle,
D - diameter, P - number "Pi".
The circumference of a circle can also be calculated through its radius (r), which is equal to half the length of the diameter. Here is the second formula you need to remember:
How to find out the diameter of a circle?
It is a chord that passes through the center of the figure. At the same time, it connects the two most distant points in the circle. Based on this, you can independently draw the diameter (radius) and measure its length using a ruler.
Method 1: enter right triangle in a circle
Calculating the circumference of a circle will be easy if we find its diameter. It is necessary to draw in a circle where the hypotenuse will be equal to the diameter of the circle. To do this, you need to have a ruler and a square on hand, otherwise nothing will work.
Method 2: fit any triangle
On the side of the circle we mark any three points, connect them - we get a triangle. It is important that the center of the circle lies in the area of the triangle; this can be done by eye. We draw medians to each side of the triangle, the point of their intersection coincides with the center of the circle. And when we know the center, we can easily draw the diameter using a ruler.
This method is very similar to the first, but can be used in the absence of a square or in cases where it is not possible to draw on a figure, for example on a plate. You need to take a sheet of paper with right angles. We apply the sheet to the circle so that one vertex of its corner touches the edge of the circle. Next, mark with dots the places where the sides of the paper intersect with the circle line. Connect these points using a pencil and ruler. If you don't have anything at hand, just fold the paper. This line will be equal to the length of the diameter.
Sample task
- We look for the diameter using a square, ruler and pencil according to method No. 1. Let's assume it turns out to be 5 cm.
- Knowing the diameter, we can easily insert it into our formula: P = d P = 5 * 3.14 = 15.7 In our case, it turned out to be about 15.7. Now you can easily explain how to calculate the circumference of a circle.
Class students secondary schools in the course they study the circle and circle as a geometric figure, and everything connected with this figure. The children become familiar with concepts such as radius and diameter, circumference or perimeter, area of a circle. It is on this topic that they learn about the mysterious number Pi - this is the Ludolph number, as it was called before. The number Pi is irrational, since its representation in the form decimal endlessly. In practice, its truncated version of three numbers is used: 3.14. This constant expresses the ratio of the length of any circle to its diameter.
Sixth-graders solve problems by deducing, from the same data and the number “Pi,” the remaining characteristics of a circle and a circle. In notebooks and on the chalkboard, they draw abstract spheres to scale and make meaningless calculations.
But in practice
In practice, such a problem may arise in a situation where, for example, there is a need to lay out a course of a certain length to hold some competition with the start and finish in one place. Having calculated the radius, you can select the passage of this route on the plan, with a compass in hand, considering the options, taking into account geographical features region. By moving the leg of the compass - the equidistant center from the future route, it is possible already at this stage to foresee where in the sections there will be ascents and where there will be descents, taking into account the natural differences in the relief. You can also immediately decide on the areas where it is best to place stands for fans.
Radius from a circle
So, let's assume that for an autocross competition you need a circular track 10,000 m long. Here is the necessary formula to determine the radius (R) of a circle given its known length (C):
R=C/2п (п – number equal to 3.14).
By substituting the available values, you can easily get the result:
R = 10,000:3.14 = 3,184.71 (m) or 3 km 184 m and 71 cm.
From radius to area
Knowing the radius of the circle, you can easily determine the area that will be removed from the landscape. Formula for area of a circle (S): S=пR2
At R = 3,184.71 m, it will be: S = 3.14 x 3,184.71 x 3,184.71 = 31,847,063 (sq. m) or almost 32 square kilometers.
Similar calculations can be useful when fencing. For example, you have enough material for a fence. Taking this value as the perimeter of the circle, you can easily determine its diameter (radius) and area, and, therefore, visually imagine the size of the future fenced area.
A circle consists of many points that are at equal distances from the center. It's flat geometric figure, and finding its length is not difficult. A person encounters a circle and a circle every day, regardless of what field he works in. Many vegetables and fruits, devices and mechanisms, dishes and furniture are round in shape. A circle is the set of points that lies within the boundaries of the 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. Interior 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 is not equal to unity, the ratio AX/BX. In a circle, this condition must be met; otherwise, this figure does not have the shape of a circle. Each point that makes up a figure is subject to the following rule: the sum of the squared distances from these points to the other two 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 relating to it. The main parameters of the figure are diameter, radius and chord. The radius is the segment connecting the center of the circle with any point on its curve. The magnitude of a chord is equal to the distance between two points on the curve of the 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 dimensions of a circle:
Diameter in calculation formulas
In economics and mathematics there is often a need to find the circumference of a circle. But also in everyday life You may encounter this need, for example, when building a fence around a round pool. How to calculate the circumference of a circle by diameter? In this case, use the formula C = 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 required value (in this example, the length of the fence): 3.14*50 = 157 meters. If the fence posts stand at a distance of three meters from each other, then a total of 52 of them will be needed.
Radius calculations
How to calculate the circumference of a circle from a known radius? To do this, use the formula C = 2*π*r, where C is the length, r is the radius. The radius in a circle is half the diameter, and this rule can be useful in everyday life. For example, in the case of preparing a pie in a sliding form.
To prevent the culinary product from getting dirty, it is necessary to use a decorative wrapper. How to cut a paper circle of the appropriate 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 shape is 20 centimeters, respectively, its radius is 10 centimeters. Using these parameters, the required circle size is found: 2*10*3, 14 = 62.8 centimeters.
Handy calculation methods
If it is not possible to find the circumference using the formula, then you should use available methods for calculating this value:
- If a round object is small, its length can be found using a rope wrapped around it once.
- The size of a large object is measured as follows: a rope is laid out on a flat surface, and a circle is rolled along it once.
- Modern students and schoolchildren use calculators for calculations. Online, you can find out unknown quantities using known parameters.
Round objects in the history of human life
The first round-shaped product that man invented was the wheel. The first structures were small round logs mounted on an axle. Then came wheels made of wooden spokes and rims. Gradually, metal parts were added to the product to reduce wear. It was in order to find out the length of the metal strips for the wheel upholstery that scientists of past centuries were looking for a formula for calculating this value.
Has a wheel shape potter's wheel , most parts in complex mechanisms, designs of water mills and spinning wheels. Round objects are often found in construction - frames of round windows in Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in their field professional activities are faced with the need to calculate the size of a circle.