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 also in Everyday life you may 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 students 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 added to the product metal parts 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.

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, which is denoted by the Greek letter π .

Determining the circumference of a circle

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

A task:

Calculate circumference with a radius of 10 centimeters.

Formula for calculating the dyne of a circle looks like:

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 circumference of a circle with a radius of 10 centimeters is:

L = 2 × 3.14 × 10 = 62.8 centimeters

Circle is a geometric figure, which is a collection of all points on the plane, remote from a given point, which is called its center, at a distance that is not equal to zero and is called the radius. Scientists knew how 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 of a circle was compiled around 1900 BC in ancient Babylon.

With such geometric shapes like circles we collide daily and everywhere. It is its shape that has the outer surface of the wheels, which are equipped with various vehicles. This detail, despite its outward simplicity and unpretentiousness, is considered one of the greatest inventions of mankind, and it is interesting that the natives of Australia and the American Indians, until the arrival of the 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 improved, their design became more and more complex, and for their manufacture it was necessary to use 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 upholster it with metal strips. In order to determine the lengths of these elements, it is necessary to use the formula for calculating the circumference (although in practice, most likely, the craftsmen did this “by eye” or simply girding the wheel with a strip and cutting off the required section of it).

It should be noted that wheel is used not only in vehicles. For example, a potter's wheel has its shape, as well as elements of gears of gears widely used in technology. Since ancient times, wheels have 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 used to make threads from animal wool and plant fibers.

circles often found in construction. Their shape is quite 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 a special tool. One of the varieties of round windows are portholes installed in ships and aircraft.

Thus, to solve the problem of determining the circumference of a circle, it is often necessary for design engineers who develop various machines, mechanisms and units, as well as architects and designers. Since the number π necessary for this is infinite, then it is not possible to determine this parameter with absolute accuracy, and therefore, the calculations take into account that degree of it, which in a particular case is necessary and sufficient.

And what is its difference from the circle. Take a pen or colors and draw a regular circle on a piece of paper. Paint over the entire middle of the resulting figure with a blue pencil. The red outline denoting the boundaries of the figure is a circle. But the blue content inside it is the circle.

The dimensions of a circle and a circle are determined by the diameter. On the red line denoting the circle, mark two points so that they are mirror images of each other. Connect them with a line. The segment must pass through the point at the center of the circle. This segment, connecting the opposite parts of the circle, is called the diameter in geometry.

A segment that does not extend through the center of the circle, but merges with it at opposite ends, is called a chord. Therefore, the chord passing through the point of the center of the circle is its diameter.

The diameter is denoted by the Latin letter D. You can find the diameter of a circle by such values ​​as the area, length and radius of the circle.

The distance from the center point to the point plotted on the circle is called the radius and is denoted by the letter R. Knowing the value of the radius helps to calculate the diameter of the circle in one simple step:

For example, the radius is 7 cm. We multiply 7 cm by 2 and get a value equal to 14 cm. Answer: D of a given figure is 14 cm.

Sometimes it is necessary to determine the diameter of a circle only by its length. Here it is necessary to apply a special formula to help determine the Formula L \u003d 2 Pi * R, where 2 is a constant value (constant), and Pi \u003d 3.14. And since it is known that R \u003d D * 2, the formula can be represented in another way

This expression is also applicable as a formula for the diameter of a circle. Substituting the known values ​​in the problem, we solve the equation with one unknown. Let's say the length is 7 m. Therefore:

Answer: The diameter is 21.98 meters.

If the value of the area is known, then the diameter of the circle can also be determined. The formula that applies in this case looks like this:

D = 2 * (S / Pi) * (1 / 2)

S - in this case Let's say in the problem it is equal to 30 square meters. m. We get:

D=2*(30/3.14)*(1/2) D=9.55414

When the value indicated in the problem is equal to the volume (V) of the ball, the following formula for finding the diameter is applied: D = (6 V / Pi) * 1/3.

Sometimes you have to find the diameter of a circle inscribed in a triangle. To do this, by the formula we find the radius of the presented circle:

R = S / p (S is the area of ​​the given triangle and p is the perimeter divided by 2).

The result is doubled, given that D = 2 * R.

It is often necessary to find the diameter of a circle in everyday life. For example, when determining what is equivalent to its diameter. To do this, wrap the finger of the potential owner of the ring with a thread. Mark the points of contact between the two ends. Measure the length from point to point with a ruler. The resulting value is multiplied by 3.14, following the formula for determining the diameter with a known length. So, the statement that knowledge in geometry and algebra will not be useful in life does not always correspond to reality. And this is a serious reason to treat school subjects more responsibly.

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:

1. Using degrees: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
2. 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 common points, she 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 magnitudes 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

The circle calculator is a service specially designed to calculate 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 sphere, but you need to get its area. There is nothing 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 value of the radius 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 such a scheme. Regardless of which initial parameter you choose, the radius value is calculated first and all subsequent calculations are based on it. For greater accuracy of calculations, the site uses the number Pi rounded to the 10th decimal place.

Calculate Diameter

Diameter calculation is the simplest type of calculation that our calculator can perform. Getting the diameter value is not difficult at all and manually, for this you do not need to resort to the help of the Internet at all. The diameter is equal to the value of the radius multiplied by 2. The diameter is the most important parameter of the circle, which is extremely often used in everyday life. Absolutely everyone should be able to calculate it correctly and use it. Using the capabilities of our site, you will calculate the diameter with great accuracy in a fraction of a second.

Find out the circumference of a circle

You can't even imagine how many round objects 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 carried out both on a piece of paper and with the help of this Internet assistant. The advantage of the latter is that it will illustrate all the calculations with drawings. And to everything else, the second method is much faster.

Calculate the area of ​​a circle

The area of ​​the circle - like all the parameters listed in this article, is the basis of modern civilization. To be 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 an area 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 the area of ​​any circle effortlessly. 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 giving people the ability to accurately calculate the area of ​​a sphere for many years. Where can it 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 ball is too wide.

Calculate the volume of a sphere

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 site makes it possible to calculate the volume of a ball in seconds, if you know any of the following options: radius, diameter, circumference, area of ​​a circle, or area of ​​a sphere. You can also use it for inverse calculations, for example, to know the volume of a ball, get the value of its radius or diameter. Thank you for briefly reviewing the capabilities of our lap calculator. We hope you enjoyed your stay with us and have already added the site to your bookmarks.