The radius is 6 what is the diameter. How to find and what will be the circumference of a circle

A circle is made up of many points that are equidistant from the center. This is a flat geometric figure, and finding its length is not difficult. A person encounters a circle and a circle every day, regardless of the area in which he works. Many vegetables and fruits, devices and mechanisms, dishes and furniture have a round shape. A circle is a set of points that is within the boundaries of a circle. Therefore, the length of the figure is equal to the perimeter of the circle.

Characteristics of the figure

In addition to the fact that the description of the concept of a circle is quite simple, its characteristics are also easy to understand. With their help, you can calculate its length. The inner part of the circle consists of many points, among which two - A and B - can be seen at right angles. This segment is called the diameter, it consists of two radii.

Within the circle there are points X such, which does not change and does not equal unity, the ratio AX / BX. In a circle, this condition is necessarily observed, otherwise this figure does not have the shape of a circle. The rule applies to each point that makes up the figure: the sum of the squared distances from these points to two others always exceeds half the length of the segment between them.

Basic circle terms

In order to be able to find the length of a figure, you need to know the basic terms related to it. The main parameters of the figure are diameter, radius and chord. A radius is a segment that connects the center of a circle with any point on its curve. The value of a chord is equal to the distance between two points on the curved figure. Diameter - distance between points passing through the center of the figure.

Basic formulas for calculations

The parameters are used in the formulas for calculating the values of the circle:

Diameter in calculation formulas

In economics and mathematics, it often becomes necessary to find the circumference of a circle. But in everyday life, you can also encounter this need, for example, during the construction of a fence around a round pool. How to calculate the circumference of a circle from a diameter? In this case, use the formula C \u003d π * D, where C is the desired value, D is the diameter.

For example, the width of the pool is 30 meters, and the fence posts are planned to be placed at a distance of ten meters from it. In this case, the formula for calculating the diameter is: 30+10*2 = 50 meters. The desired value (in this example, the length of the fence): 3.14 * 50 \u003d 157 meters. If the fence posts stand at a distance of three meters from each other, then a total of 52 will be needed.

Radius calculations

How to calculate the circumference of a circle from a known radius? For this, the formula C \u003d 2 * π * r is used, where C is the length, r is the radius. The radius in a circle is less than half the diameter, and this rule can come in handy in everyday life. For example, in the case of making a pie in a sliding form.

In order for the culinary product not to get dirty, it is necessary to use a decorative wrapper. And how to cut a paper circle of a suitable size?

Those who are a little familiar with mathematics understand that in this case you need to multiply the number π by twice the radius of the shape used. For example, the diameter of the mold is 20 centimeters, respectively, its radius is 10 centimeters. According to these parameters, the required circle size is found: 2 * 10 * 3, 14 \u003d 62.8 centimeters.

Handy calculation methods

If it is not possible to find the circumference using the formula, then you should use the available methods for calculating this value:

With a small round object, its length can be found using a rope wrapped around once.
The size of a large object is measured as follows: a rope is laid out on a flat plane, and a circle is rolled over it once.
Modern students and schoolchildren use calculators for calculations. Known parameters can be used to find out unknown values online.

Round objects in the history of human life

The first round product that man invented was the wheel. The first structures were small rounded logs mounted on axles. Then came wheels made of wooden spokes and rims. Gradually, metal parts were added to the product to reduce wear. It was in order to find out the length of the metal strips for the upholstery of the wheel that scientists of past centuries were looking for a formula for calculating this value.

The potter's wheel is shaped like a wheel, most of the details in complex mechanisms, designs of water mills and spinning wheels. Often there are round objects in construction - the frames of round windows in the Romanesque architectural style, portholes in ships. Architects, engineers, scientists, mechanics and designers every day in the field of their professional activities are faced with the need to calculate the size of a circle.

Let's first understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. This is an infinite number of points in the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that bounds it (o-circle (g)ness), and an uncountable number of points that are inside the circle.

For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).

A line segment that connects two points on a circle is chord.

A chord passing directly through the center of a circle is diameter this circle (D) . The diameter can be calculated using the formula: D=2R

Circumference calculated by the formula: C=2\pi R

Area of a circle: S=\pi R^(2)

arc of a circle called that part of it, which is located between two of its points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. The same chords subtend the same arcs.

Central corner is the angle between two radii.

arc length can be found using the formula:

Using degrees: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
Using a radian measure: CD = \alpha R

The diameter that is perpendicular to the chord bisects the chord and the arcs it spans.

If the chords AB and CD of the circle intersect at the point N, then the products of the segments of the chords separated by the point N are equal to each other.

AN\cdot NB = CN \cdot ND

Tangent to circle

Tangent to a circle It is customary to call a straight line that has one common point with a circle.

If a line has two points in common, it is called secant.

If you draw a radius at the point of contact, it will be perpendicular to the tangent to the circle.

Let's draw two tangents from this point to our circle. It turns out that the segments of the tangents will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.

AC=CB

Now we draw a tangent and a secant to the circle from our point. We get that the square of the length of the tangent segment will be equal to the product of the entire secant segment by its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an integer segment of the first secant by its outer part is equal to the product of an integer segment of the second secant by its outer part.

AC \cdot BC = EC \cdot DC

Angles in a circle

The degree measures of the central angle and the arc on which it rests are equal.

\angle COD = \cup CD = \alpha ^(\circ)

Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.

You can calculate it by knowing the size of the arc, since it is equal to half of this arc.

\angle AOB = 2 \angle ADB

Based on diameter, inscribed angle, straight.

\angle CBD = \angle CED = \angle CAD = 90^ (\circ)

Inscribed angles that lean on the same arc are identical.

The inscribed angles based on the same chord are identical or their sum equals 180^ (\circ) .

\angle ADB + \angle AKB = 180^ (\circ)

\angle ADB = \angle AEB = \angle AFB

On the same circle are the vertices of triangles with identical angles and a given base.

An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values of the arcs of the circle that are inside the given and vertical angles.

\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)

An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular magnitudes of the arcs of a circle that are inside the angle.

\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)

Inscribed circle

Inscribed circle is a circle tangent to the sides of the polygon.

At the point where the bisectors of the angles of the polygon intersect, its center is located.

A circle may not be inscribed in every polygon.

The area of a polygon with an inscribed circle is found by the formula:

S=pr,

p is the semiperimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is:

r = \frac(S)(p)

The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle is inscribed in a convex quadrilateral if the sums of the lengths of opposite sides in it are identical.

AB+DC=AD+BC

It is possible to inscribe a circle in any of the triangles. Only one single. At the point where the bisectors of the inner angles of the figure intersect, the center of this inscribed circle will lie.

The radius of the inscribed circle is calculated by the formula:

r = \frac(S)(p) ,

where p = \frac(a + b + c)(2)

Circumscribed circle

If a circle passes through every vertex of a polygon, then such a circle is called circumscribed about a polygon.

The center of the circumscribed circle will be at the point of intersection of the perpendicular bisectors of the sides of this figure.

The radius can be found by calculating it as the radius of a circle that is circumscribed about a triangle defined by any 3 vertices of the polygon.

There is the following condition: a circle can be circumscribed around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .

\angle A + \angle C = \angle B + \angle D = 180^ (\circ)

Near any triangle it is possible to describe a circle, and one and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.

The radius of the circumscribed circle can be calculated by the formulas:

R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)

R = \frac(abc)(4S)

a, b, c are the lengths of the sides of the triangle,

S is the area of the triangle.

Ptolemy's theorem

Finally, consider Ptolemy's theorem.

Ptolemy's theorem states that the product of diagonals is identical to the sum of the products of opposite sides of an inscribed quadrilateral.

AC \cdot BD = AB \cdot CD + BC \cdot AD

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.

It 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 apparent 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 running 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.

So the circumference ( C) can be calculated by multiplying the constant π per diameter ( D), or by 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 is the 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.

Problems for the circumference

Task 1. Find the circumference of a circle if its diameter is 5 cm.

Since the circumference is π multiplied by the diameter, then the circumference of a circle with a diameter of 5 cm will be equal to:

C≈ 3.14 5 = 15.7 (cm)

Task 2. Find the circumference 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 find the circumference of the circle 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 given its length, divide the circumference by 2. π

Area of a circle

The area of a circle is equal to the product of the number π to the square of the radius. The formula for finding the area of a circle:

S = pr 2

where S is the area of the circle, and r is the radius of the circle.

Since the diameter of a circle is twice the radius, the radius is equal to the diameter divided by 2:

Problems for 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 \u003d 3.14 4 \u003d 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 we calculate the area of the circle using the formula:

S = pr 2 ≈ 3.14 3.5 2 \u003d 3.14 12.25 \u003d 38.465 (cm 2)

This problem can be solved in another way. Instead of first finding the radius, you can use the formula for finding the area of a circle in terms of the diameter:

S = π	D 2	≈ 3,14	7 2	= 3,14	49	=	153,86	\u003d 38.465 (cm 2)
	4		4		4		4

Task 3. Find the radius of the circle if its area is 12.56 m 2.

To find the radius of a circle given its area, divide the area of the circle π , and then take the square root of the result:

r = √S : π

so the radius will be:

r≈ √12.56: 3.14 = √4 = 2 (m)

Number π

The circumference of objects surrounding us can be measured using a centimeter tape or a rope (thread), the length of which can then be measured separately. But in some cases it is difficult or almost impossible to measure the circumference, for example, the inner circumference of a bottle or just the circumference 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 a few round objects, from which you can measure both the circumference and the diameter. We calculate the ratio of length to diameter, as a result we get the following series of numbers:

From this we can conclude that the ratio of the circumference 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 circumference of a circle with a radius of 3 cm, you need to multiply the radius by 2 (so we get the diameter), and multiply the resulting diameter by π . Finally, with the number π we learned that the circumference of a circle with a radius of 3 cm is 18.84 cm.