The circle occurs at Everyday life no less often than a rectangle. And for many people, the problem of how to calculate the circumference is difficult. And all because it has no corners. If they were available, everything would become much easier.
What is a circle and where does it occur?
This flat figure represents a number of points that are located at the same distance from another one, which is the center. This distance is called the radius.
In everyday life, it is not often necessary to calculate the circumference of a circle, except for people who are engineers and designers. They create designs for mechanisms that use, for example, gears, portholes and wheels. Architects create houses with round or arched windows.
Each of these and other cases requires its own precision. Moreover, it turns out to be impossible to calculate the circumference absolutely accurately. This is due to the infinity of the main number in the formula. "Pi" is still being refined. And the rounded value is most often used. The degree of accuracy is chosen to give the most correct answer.
Designations of quantities and formulas
Now it’s easy to answer the question of how to calculate the circumference of a circle by radius; for this you will need the following formula:
Since radius and diameter are related to each other, there is another formula for calculations. Since the radius is two times smaller, the expression will change slightly. And the formula for how to calculate the circumference of a circle, knowing the diameter, will be as follows:
l = π * d.
What if you need to calculate the perimeter of a circle?
Just remember that a circle includes all the points inside the circle. This means that its perimeter coincides with its length. And after calculating the circumference, put an equal sign with the perimeter of the circle.
By the way, their designations are the same. This applies to radius and diameter, and the perimeter is the Latin letter P.
Examples of tasks
Task one
Condition. Find out the length of a circle whose radius is 5 cm.
Solution. Here it is not difficult to understand how to calculate the circumference. You just need to use the first formula. Since the radius is known, all you need to do is substitute the values and calculate. 2 multiplied by a radius of 5 cm gives 10. All that remains is to multiply it by the value of π. 3.14 * 10 = 31.4 (cm).
Answer: l = 31.4 cm.
Task two
Condition. There is a wheel whose circumference is known and equal to 1256 mm. It is necessary to calculate its radius.
Solution. In this task you will need to use the same formula. But only the known length will need to be divided into the product of 2 and π. It turns out that the product will give the result: 6.28. After division, the number left is: 200. This is the desired value.
Answer: r = 200 mm.
Task three
Condition. Calculate the diameter if the circumference of the circle is known, which is 56.52 cm.
Solution. Similar to the previous problem, you will need to divide the known length by the value of π, rounded to the nearest hundredth. As a result of this action, the number 18 is obtained. The result is obtained.
Answer: d = 18 cm.
Problem four
Condition. The clock hands are 3 and 5 cm long. You need to calculate the lengths of the circles that describe their ends.
Solution. Since the arrows coincide with the radii of the circles, the first formula is required. You need to use it twice.
For the first length, the product will consist of factors: 2; 3.14 and 3. The result will be 18.84 cm.
For the second answer, you need to multiply 2, π and 5. The product will give the number: 31.4 cm.
Answer: l 1 = 18.84 cm, l 2 = 31.4 cm.
Task five
Condition. A squirrel runs in a wheel with a diameter of 2 m. How far does it run in one full revolution of the wheel?
Solution. This distance is equal to the circumference. Therefore, you need to use a suitable formula. Namely, multiply the value of π and 2 m. Calculations give the result: 6.28 m.
Answer: The squirrel runs 6.28 m.
Many objects in the world around us are round in 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 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. A circle is an area within a circle. The perimeter of a circle is its length. According to different 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 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, 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! Various mnemonic rules have been invented to remember the first few digits of the number π. 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 that will help you 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 be useful. 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.
Obviously, the boundary of any circle is a circle. Therefore, the concept of the perimeter of a circle coincides with the concept of circumference. Therefore, first let us remember what a circle is and what concepts are associated with it.
Circle concept
Definition 1
We will call this a circle geometric figure, which will consist of all such points that are at the same distance from any given point.
Definition 2
We will call the center of the circle the point that is specified within Definition 1.
Definition 3
The radius of a circle will be the distance from the center of this circle to any of its points (Fig. 1).
In the Cartesian coordinate system $xOy$ we can also introduce the equation of any circle. Let us denote the center of the circle by 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 our chosen center. 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 $n$-gon 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 find that the relation $\frac(ρ)(ρ")=\frac(2τ)(2τ")$ will be true regardless of the value 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 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 $3.14$ (there is no exact value for this number, 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
Sample tasks
Example 1
Find the perimeter of a circle that is inscribed in a square with side equal to $α$.
Let us be given a square $ABCD$ into which a circle with center $O$ is inscribed. Let's draw a picture according to the conditions of the problem (Fig. 3).
Obviously, the center of the circle will coincide with the center of the square in which it is inscribed. Since the square is circumscribed around a circle, its sides will be tangent to it, that is, the radius drawn, for example, to side $AB$ will be perpendicular to it. This means that the diameter of the circle is equal to the side of the square. That is
$τ=\frac(α)(2)$
Using the formula for the perimeter of a circle, we get that
$C=2π\cdot \frac(α)(2)=πα$
Answer: $πα$.
Example 2
Find the perimeter of the circle that is described by a right triangle with legs equal to $α$ and $β$.
Let us be given a triangle $ABC$ with a right angle $C$, which has a circumscribed circle with center $O$. As we know, the diameter of such a circle is the hypotenuse of such a triangle. That is, $|AO|=|OB|=|OC|=τ$ (Fig. 4).
According to the Pythagorean theorem, the hypotenuse is equal to
$|AB|=\sqrt(α^2+β^2)$
$|AO|=τ=\frac(\sqrt(α^2+β^2))(2)$
The perimeter of a circle, according to the formula, is equal to
$C=2π\cdot \frac(\sqrt(α^2+β^2))(2)=π\sqrt(α^2+β^2)$
Answer: $π\sqrt(α^2+β^2)$.
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 value was named from this letter in Ancient Greek 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, we 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 on 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.
Circle calculator is a service specially designed for calculating the geometric dimensions of shapes online. Thanks to this service, you can easily determine any parameter of a figure based on a circle. For example: You know the volume of a ball, but you need to get its area. Nothing could be easier! Select the appropriate option, enter a numeric value, and click the Calculate button. The service not only displays the results of calculations, but also provides the formulas by which they were made. Using our service, you can easily calculate the radius, diameter, circumference (perimeter of a circle), the area of a circle and a ball, and the volume of a ball.
Calculate radius
The task of calculating the radius value is one of the most common. The reason for this is quite simple, because knowing this parameter, you can easily determine the value of any other parameter of a circle or ball. Our site is built exactly on this scheme. Regardless of what initial parameter you have chosen, the radius value is first calculated and all subsequent calculations are based on it. For greater accuracy of calculations, the site uses Pi, rounded to the 10th decimal place.
Calculate diameter
Calculating diameter is the simplest type of calculation that our calculator can perform. It is not at all difficult to obtain the diameter value manually; for this you do not need to resort to the Internet at all. Diameter equal to the value radius multiplied by 2. Diameter is the most important parameter of a circle, which is extremely often used in everyday life. Absolutely everyone should be able to calculate and use it correctly. Using the capabilities of our website, you will calculate the diameter with great accuracy in a fraction of a second.
Find out the circumference
You can’t even imagine how many round objects there are around us and what an important role they play in our lives. The ability to calculate the circumference is necessary for everyone, from an ordinary driver to a leading design engineer. The formula for calculating the circumference is very simple: D=2Pr. The calculation can be easily done either on a piece of paper or using this online assistant. The advantage of the latter is that it illustrates all calculations with pictures. And on top of everything else, the second method is much faster.
Calculate the area of a circle
The area of a circle - like all the parameters listed in this article - is the basis of modern civilization. Being able to calculate and know the area of a circle is useful for all segments of the population without exception. It is difficult to imagine a field of science and technology in which it would not be necessary to know the area of a circle. The formula for calculation is again not difficult: S=PR 2. This formula and our online calculator will help you find out the area of any circle without any extra effort. Our site guarantees high accuracy of calculations and their lightning-fast execution.
Calculate the area of a sphere
The formula for calculating the area of a ball is no more complicated than the formulas described in the previous paragraphs. S=4Pr 2 . This simple set of letters and numbers has been allowing people to calculate the area of a ball quite accurately for many years. Where can this be applied? Yes everywhere! For example, you know that the area of the globe is 510,100,000 square kilometers. It is useless to list where knowledge of this formula can be applied. The scope of the formula for calculating the area of a sphere is too wide.
Calculate the volume of the ball
To calculate the volume of the ball, use the formula V = 4/3 (Pr 3). It was used to create our online service. The site website makes it possible to calculate the volume of the ball in a matter of seconds, if you know any of following parameters: radius, diameter, circumference, area of a circle or area of a sphere. You can also use it for reverse calculation, for example, to know the volume of a ball and obtain the value of its radius or diameter. Thank you for taking a quick look at the capabilities of our circle calculator. We hope you liked our site and have already bookmarked the site.