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 and is denoted by the Greek letter π .
Determining the circumference
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, having a radius of 10 centimeters.
Solution:Formula for calculating the circumference of a circle has the form:
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 length of a circle having a radius of 10 centimeters is:
L = 2 × 3.14 × 5 = 31.4 centimeters
Circle is a geometric figure, which is a collection of all points on a plane removed from a given point, which is called its center, by a certain distance not equal to zero and called the radius. Scientists were able 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 was compiled around 1900 BC in ancient Babylon.
We encounter geometric shapes such as circles every day and everywhere. It is its shape that has the outer surface of the wheels that are equipped with various vehicles. This detail, despite its apparent simplicity and unpretentiousness, is considered one of the greatest inventions of mankind, and it is interesting that the Australian aborigines and American Indians, until the arrival of 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 was improved, their design became more and more complex, and their manufacture required the use of 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 cover it with metal strips. In order to determine the lengths of these elements, it is necessary to use a formula for calculating the circumference (although in practice, most likely, the craftsmen did this “by eye” or simply by encircling the wheel with a strip and cutting off the required section).
It should be noted that wheel It is not only used in vehicles. For example, its shape is shaped like a potter's wheel, as well as elements of gears of gears, widely used in technology. Wheels have long 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, which were used to make threads from animal wool and plant fibers.
Circles can often be found in construction. Their shape is shaped by fairly 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 special tools. One of the varieties of round windows are portholes installed in ships and aircraft.
Thus, design engineers who develop various machines, mechanisms and units, as well as architects and designers, often have to solve the problem of determining the circumference of a circle. Since the number π , necessary for this, is infinite, it is not possible to determine this parameter with absolute accuracy, and therefore, in the calculations, the degree of it is taken into account, which in a particular case is necessary and sufficient.
Thus, the circumference ( C) can be calculated by multiplying the constant π per diameter ( D), or multiplying π by twice the radius, since the diameter is equal to two radii. Hence, circumference formula will look like this:
C = πD = 2πR
Where C- circumference, π - constant, D- circle diameter, R- radius of the circle.
Since a circle is the boundary of a circle, the circumference of a circle can also be called the length of a circle or the perimeter of a circle.
Circumference problems
Task 1. Find the circumference of a circle if its diameter is 5 cm.
Since the circumference is equal to π multiplied by the diameter, then the length of a circle with a diameter of 5 cm will be equal to:
C≈ 3.14 5 = 15.7 (cm)
Task 2. Find the length of a circle whose radius is 3.5 m.
First, find the diameter of the circle by multiplying the length of the radius by 2:
D= 3.5 2 = 7 (m)
Now let's find the circumference by multiplying π per diameter:
C≈ 3.14 7 = 21.98 (m)
Task 3. Find the radius of a circle whose length is 7.85 m.
To find the radius of a circle based on its length, you need to divide the circumference by 2 π
Area of a circle
The area of a circle is equal to the product of the number π per square radius. Formula for finding the area of a circle:
S = πr 2
Where S is the area of the circle, and r- radius of the circle.
Since the diameter of a circle is equal to twice the radius, the radius is equal to the diameter divided by 2:
Problems involving the area of a circle
Task 1. Find the area of a circle if its radius is 2 cm.
Since the area of a circle is π multiplied by the radius squared, then the area of a circle with a radius of 2 cm will be equal to:
S≈ 3.14 2 2 = 3.14 4 = 12.56 (cm 2)
Task 2. Find the area of a circle if its diameter is 7 cm.
First, find the radius of the circle by dividing its diameter by 2:
7:2=3.5(cm)
Now let's calculate the area of the circle using the formula:
S = πr 2 ≈ 3.14 3.5 2 = 3.14 12.25 = 38.465 (cm 2)
This problem can be solved in another way. Instead of finding the radius first, you can use the formula for finding the area of a circle using the diameter:
| S = π | D 2 | ≈ 3,14 | 7 2 | = 3,14 | 49 | = | 153,86 | = 38.465 (cm 2) |
| 4 | 4 | 4 | 4 |
Task 3. Find the radius of the circle if its area is 12.56 m2.
To find the radius of a circle from its area, you need to divide the area of the circle π , and then take the square root of the result:
r = √S : π
therefore the radius will be equal to:
r≈ √12.56: 3.14 = √4 = 2 (m)
Number π
The circumference of objects surrounding us can be measured using a measuring tape or rope (thread), the length of which can then be measured separately. But in some cases, measuring the circumference is difficult or practically impossible, for example, the inner circumference of a bottle or simply the circumference of a circle drawn on paper. In such cases, you can calculate the circumference of a circle if you know the length of its diameter or radius.
To understand how this can be done, let’s take several round objects whose circumference and diameter can be measured. Let's calculate the ratio of length to diameter, and as a result we get the following series of numbers:
From this we can conclude that the ratio of the length of a circle to its diameter is a constant value for each individual circle and for all circles as a whole. This relationship is denoted by the letter π .
Using this knowledge, you can use the radius or diameter of a circle to find its length. For example, to calculate the length of a circle with a radius of 3 cm, you need to multiply the radius by 2 (this is how we get the diameter), and multiply the resulting diameter by π . As a result, using the number π We learned that the length of a circle with a radius of 3 cm is 18.84 cm.
The circumference of a circle is indicated by the letter C and is calculated by the formula:
C = 2πR,
Where R - radius of the circle.
Derivation of the formula expressing the circumference
Path C and C’ are the lengths of circles of radii R and R’. Let us inscribe a regular n-gon into each of them and denote their perimeters by P n and P" n, and their sides by a n and a" n. Using the formula for calculating the side of a regular n-gon a n = 2R sin (180°/n) we get:
P n = n a n = n 2R sin (180°/n),
P" n = n · a" n = n · 2R" sin (180°/n).
Hence,
P n / P" n = 2R / 2R". (1)
This equality is valid for any value of n. We will now increase the number n without limit. Since P n → C, P" n → C", n → ∞, then the limit of the ratio P n / P" n is equal to C / C". On the other hand, by virtue of equality (1), this limit is equal to 2R/2R". Thus, C/C" = 2R/2R". From this equality it follows that C/2R = C"/2R", i.e. . The ratio of the circumference of a circle to its diameter is the same number for all circles. This number is usually denoted by the Greek letter π (“pi”).
From the equality C / 2R = π we obtain the formula for calculating the circumference of a circle of radius R:
C = 2πR.
Circular arc length
Since the length of the entire circle is 2πR, then the length l of an arc of 1° is equal to 2πR / 360 = πR / 180.
That's why length l of an arc of a circle with degree measure α expressed by the formula
l = (πR / 180) α.
1. Circumference. A circle is a closed flat curved line, all points of which are at equal distances from one point (O), called the center of the circle (Fig. 27).
The circle is drawn using a compass. To do this, the sharp leg of the compass is placed in the center, and the other (with a pencil) is rotated around the first until the end of the pencil draws a complete circle. The distance from the center to any point on the circle is called its radius. From the definition it follows that all radii of one circle are equal to each other.
A straight line segment (AB) connecting any two points of a circle and passing through its center is called diameter. All diameters of one circle are equal to each other; the diameter is equal to two radii.
How to find the circumference of a circle? In almost some cases, the circumference can be found by direct measurement. This can be done, for example, when measuring the circumference of relatively small objects (bucket, glass, etc.). To do this, you can use a tape measure, braid or cord.
In mathematics, the technique of indirectly determining the circumference is used. It consists of calculating using a ready-made formula, which we will now derive.
If we take several large and small round objects (coin, glass, bucket, barrel, etc.) and measure the circumference and diameter of each of them, we will get two numbers for each object (one measuring the circumference, and another is the length of the diameter). Naturally, for small objects these numbers will be small, and for large ones - large.
However, if in each of these cases we take the ratio of the two numbers obtained (circumference and diameter), then with careful measurement we will find almost the same number. Let us denote the circumference of the circle by the letter WITH, length of diameter letter D, then their ratio will look like C: D. Actual measurements are always accompanied by inevitable inaccuracies. But, having completed the indicated experiment and made the necessary calculations, we get for the ratio C: D approximately the following numbers: 3.13; 3.14; 3.15. These numbers differ very little from one another.
In mathematics, through theoretical considerations, it has been established that the desired ratio C: D never changes and it is equal to an infinite non-periodic fraction, the approximate value of which, accurate to ten thousandths, is equal to 3,1416 . This means that every circle is the same number of times longer than its diameter. This number is usually denoted by the Greek letter π (pi). Then the ratio of the circumference to the diameter will be written as follows: C: D = π . We will limit this number to only hundredths, i.e. take π = 3,14.
Let's write a formula to determine the circumference.
Because C: D= π , That
C = πD
i.e. the circumference is equal to the product of the number π per diameter.
Task 1. Find the circumference ( WITH) of a round room if its diameter is D= 5.5 m.
Taking into account the above, we must increase the diameter by 3.14 times to solve this problem:
5.5 3.14 = 17.27 (m).
Task 2. Find the radius of a wheel whose circumference is 125.6 cm.
This task is the reverse of the previous one. Let's find the wheel diameter:
125.6: 3.14 = 40 (cm).
Let us now find the radius of the wheel:
40: 2 = 20 (cm).
2. Area of a circle. To determine the area of a circle, one could draw a circle of a given radius on paper, cover it with transparent checkered paper, and then count the cells inside the circle (Fig. 28).
But this method is inconvenient for many reasons. Firstly, near the contour of the circle, a number of incomplete cells are obtained, the size of which is difficult to judge. Secondly, you cannot cover a large object (a round flower bed, a pool, a fountain, etc.) with a sheet of paper. Thirdly, having counted the cells, we still do not receive any rule that allows us to solve another similar problem. Because of this, we will act differently. Let's compare the circle with some figure familiar to us and do it as follows: cut a circle out of paper, cut it in half first along the diameter, then cut each half in half, each quarter in half, etc., until we cut the circle, for example, into 32 parts shaped like teeth (Fig. 29).
Then we fold them as shown in Figure 30, i.e., first we arrange 16 teeth in the form of a saw, and then we put 15 teeth into the resulting holes and, finally, we cut the last remaining tooth in half along the radius and attach one part to the left, the other - on right. Then you will get a figure resembling a rectangle.
The length of this figure (base) is approximately equal to the length of the semicircle, and the height is approximately equal to the radius. Then the area of such a figure can be found by multiplying the numbers expressing the length of the semicircle and the length of the radius. If we denote the area of a circle by the letter S, the circumference of a letter WITH, radius letter r, then we can write the formula for determining the area of a circle:
which reads like this: The area of a circle is equal to the length of the semicircle multiplied by the radius.
Task. Find the area of a circle whose radius is 4 cm. First find the length of the circle, then the length of the semicircle, and then multiply it by the radius.
1) Circumference WITH = π D= 3.14 8 = 25.12 (cm).
2) Length of half circle C / 2 = 25.12: 2= 12.56 (cm).
3) Area of the circle S = C / 2 r= 12.56 4 = 50.24 (sq. cm).
§ 118. Surface and volume of a cylinder.
Task 1. Find the total surface area of a cylinder whose base diameter is 20.6 cm and height 30.5 cm.
The following have a cylinder shape (Fig. 31): a bucket, a glass (not faceted), a saucepan and many other objects.
The complete surface of a cylinder (like the complete surface of a rectangular parallelepiped) consists of a lateral surface and the areas of two bases (Fig. 32).
To clearly imagine what we are talking about, you need to carefully make a model of a cylinder out of paper. If we subtract two bases from this model, i.e. two circles, and cut the side surface lengthwise and unfold it, then it will be completely clear how to calculate the total surface of the cylinder. The side surface will unfold into a rectangle, the base of which is equal to the length of the circle. Therefore, the solution to the problem will look like:
1) Circumference: 20.6 3.14 = 64.684 (cm).
2) Lateral surface area: 64.684 30.5 = 1972.862 (cm2).
3) Area of one base: 32.342 10.3 = 333.1226 (sq.cm).
4) Full cylinder surface:
1972.862 + 333.1226 + 333.1226 = 2639.1072 (sq. cm) ≈ 2639 (sq. cm).
Task 2. Find the volume of an iron barrel shaped like a cylinder with dimensions: base diameter 60 cm and height 110 cm.
To calculate the volume of a cylinder, you need to remember how we calculated the volume of a rectangular parallelepiped (it is useful to read § 61).
Our unit of volume measurement will be cubic centimeter. First you need to find out how many cubic centimeters can be placed on the base area, and then multiply the found number by the height.
To find out how many cubic centimeters can be laid on the base area, you need to calculate the base area of the cylinder. Since the base is a circle, you need to find the area of the circle. Then, to determine the volume, multiply it by the height. The solution to the problem has the form:
1) Circumference: 60 3.14 = 188.4 (cm).
2) Area of the circle: 94.2 30 = 2826 (sq. cm).
3) Cylinder volume: 2826,110 = 310,860 (cc. cm).
Answer. Barrel volume 310.86 cubic meters. dm.
If we denote the volume of a cylinder by the letter V, base area S, cylinder height H, then you can write a formula to determine the volume of a cylinder:
V = S H
which reads like this: The volume of a cylinder is equal to the area of the base multiplied by the height.
§ 119. Tables for calculating the circumference of a circle by diameter.
When solving various production problems, it is often necessary to calculate the circumference. Let's imagine a worker who produces round parts according to the diameters specified to him. Every time he knows the diameter, he must calculate the circumference. To save time and insure himself against mistakes, he turns to ready-made tables that indicate the diameters and the corresponding circumference lengths.
We will present a small part of such tables and tell you how to use them.
Let it be known that the diameter of the circle is 5 m. We look in the table in the vertical column under the letter D number 5. This is the length of the diameter. Next to this number (to the right, in the column called “Circumference”) we will see the number 15.708 (m). In exactly the same way we find that if D= 10 cm, then the circumference is 31.416 cm.
Using the same tables, you can also perform reverse calculations. If the circumference of a circle is known, then the corresponding diameter can be found in the table. Let the circumference be approximately 34.56 cm. Let us find in the table the number closest to this. This will be 34.558 (difference 0.002). The diameter corresponding to this circumference is approximately 11 cm.
The tables mentioned here are available in various reference books. In particular, they can be found in the book “Four-digit mathematical tables” by V. M. Bradis. and in the arithmetic problem book by S. A. Ponomarev and N. I. Sirneva.
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 of mathematical analysis through sums of series. The designation of this constant by the Greek letter π was first used by William Jones in 1706, and it 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! Various mnemonic rules have been invented to remember the first few digits of the number π. Some allow you to store a large number of numbers in memory, 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:
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 using simple formulas involving 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 a single number - diameter or radius, will help you solve this problem.
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