Instruction
At first it is necessary the initial data to the task. The fact is that its condition cannot be explicitly said what is the radius circles. Instead, the problem can be given the length of the diameter circles. Diameter circles a line segment that connects two opposite points circles passing through its center. Having analyzed the definitions circles, we can say that the length of the diameter is twice the length of the radius.
Now we can accept the radius circles equal to R. Then for the length circles you need to use the formula:
L = 2πR = πD, where L is the length circles, D - diameter circles, which is always 2 times the radius.
note
A circle can be inscribed in a polygon, or described around it. Moreover, if the circle is inscribed, then it will divide them in half at the points of contact with the sides of the polygon. To find the radius of an inscribed circle, you need to divide the area of the polygon by half its perimeter:
R = S/p.
If a circle is circumscribed around a triangle, then its radius is found by the following formula:
R \u003d a * b * c / 4S, where a, b, c are the sides of the given triangle, S is the area of \u200b\u200bthe triangle around which the circle is described.
If it is required to describe a circle around a quadrilateral, then this can be done subject to two conditions:
The quadrilateral must be convex.
The sum of the opposite angles of the quadrilateral should be 180°
Helpful advice
In addition to the traditional caliper, stencils can also be used to draw a circle. In modern stencils, a circle of different diameters is included. These stencils can be purchased at any stationery store.
Sources:
- How to find the circumference of a circle?
Circle - a closed curved line, all points of which are at an equal distance from one point. This point is the center of the circle, and the segment between the point on the curve and its center is called the radius of the circle.
Instruction
If a straight line is drawn through the center of a circle, then its segment between the two points of intersection of this line with the circle is called the diameter of this circle. Half the diameter, from the center to the point where the diameter intersects with the circle, is the radius
circles. If the circle is cut at an arbitrary point, straightened and measured, then the resulting value is the length of the given circle.
Draw several circles with different compass solutions. Visual comparison leads to the conclusion that a larger diameter outlines a larger circle bounded by a circle with a larger length. Therefore, there is a directly proportional relationship between the diameter of a circle and its length.
According to the physical meaning, the parameter "circumference" corresponds to, limited by a broken line. If a regular n-gon with side b is inscribed in a circle, then the perimeter of such a figure P is equal to the product of side b by the number of sides n: P \u003d b * n. Side b can be determined by the formula: b=2R*Sin (π/n), where R is the radius of the circle into which the n-gon is inscribed.
As the number of sides increases, the perimeter of the inscribed polygon will increasingly approach L. Р= b*n=2n*R*Sin (π/n)=n*D*Sin (π/n). The relationship between the circumference L and its diameter D is constant. The ratio L / D \u003d n * Sin (π / n) as the number of sides of the inscribed polygon tends to infinity tends to the number π, a constant value called “pi number” and expressed as an infinite decimal fraction. For calculations without the use of computer technology, the value π=3.14 is taken. The circumference of a circle and its diameter are related by the formula: L= πD. For a circle, divide its length by π=3.14.
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.
This 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 is easy to calculate with simple formulas involving 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.
In the process of performing construction work at home or at work, it may be necessary to measure the diameter of a pipe that is already installed in a water supply or sewerage system. It is also necessary to know this parameter at the design stage of laying engineering communications.
Hence the need to figure out how to determine the diameter of the pipe. The choice of a specific measurement method depends on the size of the object and whether the location of the pipeline is accessible.
Determining the diameter at home
Before measuring the diameter of the pipe, you need to prepare the following tools and devices:
- tape measure or standard ruler;
- calipers;
- camera - it will be used if necessary.
If the pipeline is available for measurements, and the ends of the pipes can be measured without problems, then it is enough to have a regular ruler or tape measure at your disposal. It should be borne in mind that such a method is used when minimal requirements are imposed on accuracy.
In this case, the pipe diameter is measured in the following sequence:
- Prepared tools are applied to the place where the widest part of the end of the product is located.
- Then count the number of divisions corresponding to the size of the diameter.
This method allows you to find out the parameters of the pipeline with an accuracy of several millimeters.
To measure the outer diameter of pipes with a small cross section, you can use a tool such as a caliper:
- Move apart its legs and apply to the end of the product.
- Then they need to be moved so that they are firmly pressed against the outer side of the pipe walls.
- Focusing on the scale of fixture values, they find out the required parameter.
This method of determining the diameter of the pipe gives fairly accurate results, down to tenths of a millimeter.
When the pipeline is not available for measurement and is part of an already functioning water supply or gas pipeline, proceed as follows: a caliper is applied to the pipe, to its side surface. In this way, the product is measured in cases where the length of the legs of the measuring device exceeds half the diameter of the tubular product.
Often in domestic conditions there is a need to learn how to measure the diameter of a pipe with a large cross section. There is a simple way to do this: it is enough to know the circumference of the product and the constant π, equal to 3.14.
First, using a tape measure or a piece of cord, measure the pipe in girth. Then the known values are substituted into the formula d \u003d l: π, where:
d is the diameter to be determined;
l is the length of the measured circle.
For example, the girth of the pipe is 62.8 centimeters, then d \u003d 62.8: 3.14 \u003d 20 centimeters or 200 millimeters.
There are situations when the laid pipeline is completely inaccessible. Then you can apply the copy method. Its essence lies in the fact that a measuring instrument or a small object with known parameters is applied to the pipe.
For example, it can be a box of matches, the length of which is 5 centimeters. Then this section of the pipeline is photographed. Subsequent calculations are performed on the photograph. In the picture, the visible thickness of the product is measured in millimeters. Then you need to convert all the obtained values into the real parameters of the pipe, taking into account the scale of the photography.
Measurement of diameters in production conditions
At large facilities under construction, pipes must be subjected to incoming inspection before installation begins. First of all, they check the certificates and markings applied to the pipe products.
The documentation must contain certain information regarding pipes:
- nominal dimensions;
- number and date of specifications;
- brand of metal or type of plastic;
- lot number;
- the results of the tests;
- chem. smelting analysis;
- type of heat treatment;
- results of x-ray flaw detection.
In addition, on the surface of all products at a distance of approximately 50 centimeters from one of the ends, a marking is always applied containing:
- manufacturer's name;
- melt number;
- product number and its nominal parameters;
- date of manufacture;
- carbon equivalent.
Pipe lengths under production conditions are determined by measuring wire. Also, there are no difficulties with how to measure the diameter of the pipe with a tape measure.
For products of the first class, the permissible deviation in one direction or another from the declared length is 15 millimeters. For the second class - 100 millimeters.
For pipes, the outer diameter is checked using the formula d = l: π-2Δp-0.2 mm, where, in addition to the above values:
Δр – tape measure material thickness;
0.2 mm - allowance for the fit of the tool to the surface.
Deviation of the outer diameter value from the one declared by the manufacturer is allowed:
- for products with a cross section of not more than 200 millimeters - 1.5 millimeters;
- for large pipes - 0.7%.
In the latter case, ultrasonic measuring instruments are used to check tubular products. To determine the wall thickness, calipers are used, in which the division on the scale corresponds to 0.01 mm. The minus tolerance must not exceed 5% of the nominal thickness. In this case, the curvature cannot be more than 1.5 millimeters per 1 linear meter.
From the above information, it is clear that it is not difficult to figure out how to determine the diameter of the pipe along the circumference or using simple measuring tools.
A circle is a curved line that encloses a circle. In geometry, figures are flat, so the definition refers to a two-dimensional image. It is assumed that all points of this curve are at an equal distance from the center of the circle.
The circle has several characteristics, on the basis of which the calculations associated with this geometric figure are made. These include: diameter, radius, area and circumference. These characteristics are interrelated, that is, information about at least one of the components is sufficient to calculate them. For example, knowing only the radius of a geometric figure using the formula, you can find the circumference, diameter, and its area.
- The radius of a circle is a segment inside the circle connected to its center.
- Diameter is a line segment inside a circle that connects its points and passes through the center. In fact, the diameter is two radii. This is exactly what the formula for calculating it looks like: D=2r.
- There is another component of the circle - the chord. This is a straight line that connects two points on a circle, but does not always pass through the center. So the chord that passes through it is also called the diameter.
How to find the circumference of a circle? Now let's find out.
Circumference: formula
The Latin letter p has been chosen to designate this characteristic. Archimedes also proved that the ratio of the circumference of a circle to its diameter is the same number for all circles: it is the number π, which is approximately equal to 3.14159. The formula for calculating π looks like this: π = p/d. According to this formula, the value of p is equal to πd, that is, the circumference: p= πd. Since d (diameter) is equal to two radii, the same circumference formula can be written as p=2πr. Let's consider the application of the formula using simple problems as an example:
Task 1
At the base of the Tsar Bell, the diameter is 6.6 meters. What is the circumference of the base of the bell?
- So, the formula for calculating the circle is p= πd
- We substitute the existing value in the formula: p \u003d 3.14 * 6.6 \u003d 20.724
Answer: The circumference of the base of the bell is 20.7 meters.
Task 2
An artificial satellite of the Earth rotates at a distance of 320 km from the planet. The radius of the Earth is 6370 km. What is the length of the satellite's circular orbit?
- 1. Calculate the radius of the circular orbit of the Earth satellite: 6370+320=6690 (km)
- 2. Calculate the length of the circular orbit of the satellite using the formula: P=2πr
- 3.P=2*3.14*6690=42013.2
Answer: the length of the circular orbit of the Earth's satellite is 42013.2 km.
Methods for measuring the circumference
The calculation of the circumference of a circle is not often used in practice. The reason for this is the approximate value of the number π. In everyday life, a special device is used to find the length of a circle - a curvimeter. An arbitrary reference point is marked on the circle and the device is guided from it strictly along the line until they again reach this point.
How to find the circumference of a circle? You just need to keep in mind simple formulas for calculations.
- This is a flat figure, which is a set of points equidistant from the center. All of them are at the same distance and form a circle.
A line segment that connects the center of a circle with points on its circumference is called radius. In each circle, all radii are equal to each other. A line joining two points on a circle and passing through the center is called diameter. The formula for the area of a circle is calculated using a mathematical constant - the number π ..
This is interesting : The number pi. is the ratio of the circumference of a circle to the length of its diameter and is a constant value. The value π = 3.1415926 was used after the work of L. Euler in 1737.
The area of a circle can be calculated using the constant π. and the radius of the circle. The formula for the area of a circle in terms of radius looks like this:
Consider an example of calculating the area of a circle using the radius. Let a circle with radius R = 4 cm be given. Let's find the area of the figure.
The area of our circle will be equal to 50.24 square meters. cm.
There is a formula the area of a circle through the diameter. It is also widely used to calculate the required parameters. These formulas can be used to find .
Consider an example of calculating the area of a circle through the diameter, knowing its radius. Let a circle be given with a radius R = 4 cm. First, let's find the diameter, which, as you know, is twice the radius.
Now we use the data for the example of calculating the area of a circle using the above formula:
As you can see, as a result we get the same answer as in the first calculations.
Knowledge of the standard formulas for calculating the area of a circle will help in the future to easily determine sector area and it is easy to find the missing quantities.
We already know that the formula for the area of a circle is calculated through the product of the constant value π and the square of the radius of the circle. The radius can be expressed in terms of the circumference of a circle and substitute the expression in the formula for the area of a circle in terms of the circumference:
Now we substitute this equality into the formula for calculating the area of a circle and get the formula for finding the area of \u200b\u200bthe circle, through the circumference
Consider an example of calculating the area of a circle through the circumference. Let a circle with length l = 8 cm be given. Let's substitute the value in the derived formula: 
The total area of the circle will be 5 square meters. cm.
Area of a circle circumscribed around a square
It is very easy to find the area of a circle circumscribed around a square.
This will require only the side of the square and knowledge of simple formulas. The diagonal of the square will be equal to the diagonal of the circumscribed circle. Knowing the side a, it can be found using the Pythagorean theorem: from here.
After we find the diagonal, we can calculate the radius: .
And then we substitute everything into the basic formula for the area of a circle circumscribed around a square:
