Many objects in the surrounding world have round shape. These are wheels, round window openings, pipes, various dishes and much more. You can calculate the length of a circle by knowing its diameter or radius.
There are several definitions of this geometric figure.
- This is a closed curve consisting of points that are located at the same distance from a given point.
- This is a curve consisting of points A and B, which are the ends of the segment, and all points from which A and B are visible at right angles. In this case, the segment AB is the diameter.
- For the same segment AB, this curve includes all points C such that the ratio AC/BC is constant and not equal to 1.
- This is a curve consisting of points for which the following is true: if you add the squares of the distances from one point to two given other points A and B, you get constant number, greater than 1/2 of the segment connecting A and B. This definition is derived from the Pythagorean theorem.
Note! There are other definitions. A circle is an area within a circle. The perimeter of a circle is its length. By different definitions the circle may or may not include the curve itself, which is its boundary.
Definition of a circle
Formulas
How to calculate the circumference of a circle using the radius? This is done using a simple formula:
where L is the desired value,
π is the number pi, approximately equal to 3.1413926.
Usually, to find the required value, it is enough to use π to the second digit, that is, 3.14, this will provide the required accuracy. On calculators, in particular engineering ones, there may be a button that automatically enters the value of the number π.
Designations
To find through the diameter there is the following formula:
If L is already known, the radius or diameter can be easily found out. To do this, L must be divided by 2π or π, respectively.
If a circle has already been given, you need to understand how to find the circumference from this data. The area of the circle is S = πR2. From here we find the radius: R = √(S/π). Then
L = 2πR = 2π√(S/π) = 2√(Sπ).
Calculating the area in terms of L is also easy: S = πR2 = π(L/(2π))2 = L2/(4π)
To summarize, we can say that there are three basic formulas:
- through the radius – L = 2πR;
- through diameter – L = πD;
- through the area of the circle – L = 2√(Sπ).
Pi
Without the number π it will not be possible to solve the problem under consideration. The number π was first found as the ratio of the circumference of a circle to its diameter. This was done by the ancient Babylonians, Egyptians and Indians. They found it quite accurately - their results differed from the currently known value of π by no more than 1%. The constant was approximated by such fractions as 25/8, 256/81, 339/108.
Further, the value of this constant was calculated not only from the point of view of geometry, but also from the point of view mathematical analysis through sums of series. The designation of this constant Greek letterπ 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. Using supercomputer calculations, the 10-trillionth sign of the constant was discovered in 2011.
This is interesting! To remember the first few digits of the number π, various mnemonic rules. Some allow you to store in memory big number numbers, for example, one French poem will help you remember pi up to the 126th digit.
If you need the circumference, an online calculator will help you with this. There are many such calculators; you just need to enter the radius or diameter. Some of them have both of these options, others calculate the result only through R. Some calculators can calculate the desired value with different precision, you need to specify the number of decimal places. You can also calculate the area of a circle using online calculators.
Such calculators are easy to find with any search engine. There are also mobile applications, which will help solve the problem of how to find the circumference of a circle.
Useful video: circumference
Practical use
Solving such a problem is most often necessary for engineers and architects, but in everyday life knowledge necessary formulas may also come in handy. For example, you need to wrap a paper strip around a cake baked in a mold with a diameter of 20 cm. Then it will not be difficult to find the length of this strip:
L = πD = 3.14 * 20 = 62.8 cm.
Another example: you need to build a fence around a round pool at a certain distance. If the radius of the pool is 10 m, and the fence needs to be placed at a distance of 3 m, then R for the resulting circle will be 13 m. Then its length is:
L = 2πR = 2 * 3.14 * 13 = 81.68 m.
Useful video: circle - radius, diameter, circumference
Bottom line
The perimeter of a circle can be easily calculated by simple formulas, including diameter or radius. You can also find the desired quantity through the area of a circle. Online calculators or mobile applications in which you need to enter singular– diameter or radius.
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 Cartesian system coordinates \(xOy\) 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 will 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 relation \(\frac(ρ)(ρ")=\frac(2τ)(2τ") \) 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 because it is 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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Its diameter. To do this, you just need to apply the formula for circumference. L = n DHere: L – circumference, n– number Pi, equal to 3.14, D – diameter of the circle. Rearrange the required value in the formula for the circumference of the circle left side and get: D = L/n
Let's sort it out practical problem. Suppose you need to make a cover for a round country well, which is accessible within this moment No. No, and inappropriate weather. But do you have data on length its circumference. Let's assume this is 600 cm. We substitute the values into the indicated formula: D = 600/3.14 = 191.08 cm. So, the diameter of your is 191 cm. Increase the diameter to 2, taking into account the allowance for the edges. Set the compass to a radius of 1 m (100 cm) and draw a circle.
It is convenient to draw circles of relatively large diameters at home with a compass, which can be quickly made. It's done like this. Two nails are driven into the lath at a distance from each other equal to the radius of the circle. Drive one nail shallowly into the workpiece. And use the other one, rotating the staff, as a marker.
A circle is a geometric figure on a plane that consists of all points of this plane that are at the same distance from a given point. Set point in this case it is called the center circle, and the distance at which the points circle are from its center - radius circle. Plane area bounded by a circle called a circle. There are several methods of calculation diameter circle, the choice of a specific one depends on the available initial data.
Instructions
In the simplest case, if the circle is of radius R, then it will be equal to
D = 2 * R
If radius circle is not known, but it is known, then the diameter can be calculated using the length formula circle
D = L/P, where L is length circle, P – P.
Same diameter circle can be calculated knowing the area limited by it
D = 2 * v(S/P), where S is the area of the circle, P is the number P.
Sources:
- circle diameter calculation
In the course of planimetry high school, concept circle is defined as a geometric figure consisting of all points of the plane lying at a distance of a radius from a point called its center. You can draw many segments inside a circle, in various ways connecting its points. Depending on the construction of these segments, circle can be divided into several parts different ways.
Instructions
Finally, circle can be divided by constructing segments. A segment is a part of a circle made up of a chord and an arc of a circle. In this case, a chord is a segment connecting any two points on a circle. Using segments circle can be divided into infinite set parts with or without a formation at its center.
Video on the topic
note
The figures obtained by the above methods - polygons, segments and sectors - can also be divided using appropriate methods, for example, diagonals of polygons or bisectors of angles.
A flat geometric figure is called a circle, and the line that bounds it is usually called a circle. The main property is that every point on this line is the same distance from the center of the figure. A segment with a beginning at the center of the circle and ending at any point on the circle is called a radius, and a segment connecting two points on the circle and passing through the center is called a diameter.
Instructions
Use Pi to find the length of a diameter given the known circumference. This constant expresses a constant relationship between these two parameters of the circle - regardless of the size of the circle, dividing its circumference by the length of its diameter always gives the same number. It follows from this that to find the length of the diameter, the circumference should be divided by the number Pi. As a rule, for practical calculations of the length of a diameter, accuracy to hundredths of a unit is sufficient, that is, to two decimal places, so the number Pi can be considered equal to 3.14. But since this constant is an irrational number, it has infinite number decimal places. If there is a need for more precise definition, That the right number signs for pi can be found, for example, at this link - http://www.math.com/tables/constants/pi.htm.
Given the known lengths of the sides (a and b) of a rectangle inscribed in a circle, the length of the diameter (d) can be calculated by finding the length of the diagonal of this rectangle. Since the diagonal here is the hypotenuse in right triangle, the legs of which form sides of known length, then, according to the Pythagorean theorem, the length of the diagonal, and with it the length of the diameter of the circumscribed circle, can be calculated by finding from the sum of the squares of the lengths known parties: d=√(a² + b²).
Division into several equal parts- a common task. This is how you can build regular polygon, draw a star or prepare the basis for a diagram. There are several ways to solve this interesting task.
You will need
- - a circle with a designated center (if the center is not marked, you will have to find it in any way);
- - protractor;
- - compass with stylus;
- - pencil;
- - ruler.
Instructions
The easiest way to divide circle into equal parts - using a protractor. Dividing 360° into the required number of parts, you get the angle. Start from any point on the circle - the corresponding radius will be the zero mark. Starting from there, make marks on the protractor corresponding to the calculated angle. This method is recommended if you need to divide circle by five, seven, nine, etc. parts. For example, to build regular pentagon its vertices should be located every 360/5 = 72°, that is, at 0°, 72°, 144°, 216°, 288°.
To share circle into six parts, you can use the property of a regular one - its longest diagonal is equal to twice the side. A regular hexagon is, as it were, made up of six equilateral triangles. Set the compass opening equal to the radius of the circle, and make notches with it, starting from any arbitrary point. Serifs form regular hexagon, one of the vertices of which will be at this point. By connecting the vertices through one, you will build regular triangle, inscribed in circle, that is, it is divided into three equal parts.
To share circle into four parts, start with an arbitrary diameter. Its ends will give two of the required four points. To find the rest, install a compass solution, equal to a circle. Place the compass needle on one end of the diameter and make notches outside the circle and below. Repeat the same with the other end of the diameter. Draw an auxiliary line between the intersection points of the serifs. It will give you a second diameter, strictly perpendicular to the original one. Its ends will become the remaining two vertices of the square inscribed in circle.
Using the method described above, you can find the middle of any segment. As a consequence, with this method you can double the number of equal parts into which you circle. Having found the midpoint of each side of the correct n- inscribed in circle, you can draw perpendiculars to them, find the point of their intersection with circle yu and thus construct the vertices of a regular 2n-gon. This procedure can be repeated as many times as you like. So, the square turns into, that - into, etc. Starting with a square, you can, for example, divide circle into 256 equal parts.
note
To divide a circle into equal parts, dividing heads or dividing tables are usually used, which make it possible to divide the circle into equal parts with high accuracy. When it is necessary to divide a circle into equal parts, use the table below. To do this you need to multiply the diameter divisible circle by the coefficient given in the table: K x D.
Helpful advice
Dividing a circle into three, six and twelve equal parts. Carry out two perpendicular to the axis, which intersecting the circle at points 1,2,3,4 divide it into four equal parts; Using the well-known division technique right angle Using a compass or square, bisectors of right angles are constructed into two equal parts, which, intersecting with the circle at points 5, 6, 7, and 8, divide each fourth part of the circle in half.
When constructing various geometric shapes, it is sometimes necessary to determine their characteristics: length, width, height, and so on. If we're talking about about a circle or circle, you often have to determine its diameter. A diameter is a straight line segment that connects the two points furthest from each other located on a circle.
You will need
- - yardstick;
- - compass;
- - calculator.
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. Diameter of this figure equal to length 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.
- Consists of points, for each of which the following equality holds: the sum of the squares of the distances to the other two is set value, which is always more than half the length of the segment between them.
Terminology
Not everyone at school had good teacher mathematics. Therefore, the answer to the question of how to find the circumference 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 in the following way: C = π*D.
- Radius equal to half 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
For simplicity of 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 only have 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 amount 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 the 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.
Sample problems
We have already looked at several practical cases of the knowledge gained 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 the problem increased complexity. 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 similar tasks. In addition, for such inattention you can receive 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!
Very often when deciding school assignments in physics, the question arises - how to find the circumference of a circle, knowing the diameter? In fact, there are no difficulties in solving this problem; you just need to clearly imagine what formulas,concepts and definitions are required for this.
In contact with
Basic concepts and definitions
- Radius is the line connecting the center of the circle and its arbitrary point. It is designated Latin letter r.
- A chord is a line connecting two arbitrary points lying on a circle.
- Diameter is the line connecting two points of a circle and passing through its center. It is denoted by the Latin letter d.
- is a line consisting of all points located on equal distance from one selected point called its center. We will denote its length by the Latin letter l.
The area of a circle is the entire territory enclosed within a circle. It is measured V square units and is denoted by the Latin letter s.
Using our definitions, we come to the conclusion that the diameter of a circle is equal to its largest chord.
Attention! From the definition of what the radius of a circle is, you can find out what the diameter of a circle is. These are two radii laid out in opposite directions!
Diameter of a circle.
Finding the circumference and area of a circle
If we are given the radius of a circle, then the diameter of the circle is described by the formula d = 2*r. Thus, to answer the question of how to find the diameter of a circle, knowing its radius, the last one is enough multiply by two.
The formula for the circumference of a circle, expressed in terms of its radius, has the form l = 2*P*r.
Attention! The Latin letter P (Pi) denotes the ratio of the circumference of a circle to its diameter, and this is a non-periodic decimal. IN school math it is considered a previously known tabular value equal to 3.14!
Now let's rewrite the previous formula to find the circumference of a circle through its diameter, remembering what its difference is in relation to the radius. It will turn out: l = 2*P*r = 2*r*P = P*d.
From the mathematics course we know that the formula describing the area of a circle has the form: s = П*r^2.
Now let's rewrite the previous formula to find the area of a circle through its diameter. We get,
s = П*r^2 = П*d^2/4.
One of the most difficult tasks in this topic is determining the area of a circle through the circumference and vice versa. Let's take advantage of the fact that s = П*r^2 and l = 2*П*r. From here we get r = l/(2*П). Let's substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4P). In a completely similar way, the circumference is determined through the area of the circle.
Determining radius length and diameter
Important! First of all, let's learn how to measure the diameter. It's very simple - draw any radius, extend it by the opposite side until it intersects with the arc. We measure the resulting distance with a compass and use any metric tool to find out what we are looking for!
Let us answer the question of how to find out the diameter of a circle, knowing its length. To do this, we express it from the formula l = П*d. We get d = l/P.
We already know how to find its diameter from the circumference of a circle, and we can also find its radius in the same way.
l = 2*P*r, hence r = l/2*P. In general, to find out the radius, it must be expressed in terms of the diameter and vice versa.
Suppose now you need to determine the diameter, knowing the area of the circle. We use the fact that s = П*d^2/4. Let us express d from here. It will work out d^2 = 4*s/P. To determine the diameter itself, you will need to extract square root of the right side. It turns out d = 2*sqrt(s/P).
Solving typical tasks
- Let's find out how to find the diameter if the circumference is given. Let it be equal to 778.72 kilometers. Required to find d. d = 778.72/3.14 = 248 kilometers. Let's remember what a diameter is and immediately determine the radius; to do this, we divide the value d determined above in half. It will work out r = 248/2 = 124 kilometer
- Let's consider how to find the length of a given circle, knowing its radius. Let r have a value of 8 dm 7 cm. Let's convert all this into centimeters, then r will be equal to 87 centimeters. Let's use the formula to find the unknown length of a circle. Then our desired value will be equal to l = 2*3.14*87 = 546.36 cm. Let's convert our obtained value into integer numbers of metric quantities l = 546.36 cm = 5 m 4 dm 6 cm 3.6 mm.
- Let us need to determine the area of a given circle using the formula through its known diameter. Let d = 815 meters. Let's remember the formula for finding the area of a circle. Let's substitute the values given to us here, we get s = 3.14*815^2/4 = 521416.625 sq. m.
- Now we will learn how to find the area of a circle, knowing the length of its radius. Let the radius be 38 cm. We use the formula known to us. Let us substitute here the value given to us by condition. You get the following: s = 3.14*38^2 = 4534.16 sq. cm.
- The last task is to determine the area of a circle based on the known circumference. Let l = 47 meters. s = 47^2/(4P) = 2209/12.56 = 175.87 sq. m.
Circumference