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How to determine the radius knowing the circumference. How to find and what will be the circumference of a circle?

Instructions

Remember that Archimedes was the first to mathematically calculate this relationship. It is a regular 96-sided triangle in and around a circle. The perimeter of the inscribed polygon was taken to be the minimum possible circumference, and the perimeter of the circumscribed figure was taken to be the maximum size. According to Archimedes, the ratio of circumference to diameter is 3.1419. Much later, this number was “extended” to eight characters by the Chinese mathematician Zu Chongzhi. His calculations remained the most accurate for 900 years. Only in the 18th century were one hundred decimal places counted. And since 1706 this endless decimal Thanks to William Jones it acquired a name. He designated it with the first letter Greek words perimeter (periphery). Today the computer easily calculates the digits of Pi: 3.141592653589793238462643…

For calculations, reduce Pi to 3.14. It turns out that for any circle its length divided by diameter is equal to this number: L: d = 3.14.

Express from this statement a formula for finding the diameter. It turns out that to find the diameter of a circle, you need to divide the circumference by the number Pi. It looks like this: d = L: 3.14. This is a universal way to find the diameter when the circumference of a circle is known.

So, the circumference is known, say 15.7 cm, divide this figure by 3.14. The diameter will be 5 cm. Write it like this: d = 15.7: 3.14 = 5 cm.

Find the diameter from the circumference using special tables for calculating circumference. These tables are included in various reference books. For example, they are in “Four-digit mathematical tables” by V.M. Bradis.

Helpful advice

Remember the first eight digits of Pi with the help of a poem:
You just have to try
And remember everything as it is:
Three, fourteen, fifteen,
Ninety two and six.

Sources:

The number "Pi" is calculated with record accuracy
diameter and circumference
How to find the circumference of a circle?

Circle is flat geometric figure, all points of which are at the same and non-zero distance from the selected point, which is called the center of the circle. A straight line connecting any two points of a circle and passing through the center is called diameter. The total length of all the boundaries of a two-dimensional figure, which is usually called the perimeter, is more often referred to as the “circumference” of a circle. Knowing the circumference of a circle, you can calculate its diameter.

Instructions

To find the diameter, use one of the main properties of a circle, which is that the ratio of the length of its perimeter to the diameter is the same for absolutely all circles. Of course, constancy did not go unnoticed by mathematicians, and this proportion has long received its own - this is the number Pi (π is the first Greek word " circle" and "perimeter"). The numerical value of this is determined by the length of a circle whose diameter is equal to one.

Divide the known circumference of a circle by Pi to calculate its diameter. Since this number is "", it does not have final value- this is a fraction. Round Pi according to the accuracy of the result you need to obtain.

Video on the topic

Tip 4: How to find the ratio of the circumference to the diameter

Amazing property circle discovered to us by the ancient Greek scientist Archimedes. It lies in the fact that attitude her length to the diameter length is the same for any circle. In his work “On the Measurement of a Circle,” he calculated it and designated it as the number “Pi.” It is irrational, that is, its meaning cannot be accurately expressed. For this purpose its value is equal to 3.14. You can check Archimedes' statement yourself by doing simple calculations.

You will need

- compass;
- ruler;
- pencil;
- thread.

Instructions

Draw a circle of arbitrary diameter on paper with a compass. Using a ruler and pencil, draw a segment through its center connecting two lines on the line circle. Use a ruler to measure the length of the resulting segment. Let's say circle V in this case 7 centimeters.

Take the thread and arrange it along the length circle. Measure the resulting length of thread. Let it be equal to 22 centimeters. Find attitude length circle to the length of its diameter - 22 cm: 7 cm = 3.1428.... Round the resulting number (3.14). The result is the familiar number “Pi”.

Prove this property circle you can using a cup or glass. Measure their diameter with a ruler. Wrap a thread around the top of the dish and measure the resulting length. Dividing the length circle cup by the length of its diameter, you will also get the number “Pi”, making sure of this property circle, discovered by Archimedes.

Using this property you can calculate the length of any circle along the length of its diameter or according to the formulas: C = 2*p*R or C = D*p, where C - circle, D is the length of its diameter, R is the length of its radius. To find (the plane, limited by lines circle) use the formula S = π*R² if its radius is known, or the formula S = π*D²/4 if its diameter is known.

note

Did you know that Pi Day has been celebrated on the fourteenth of March for more than twenty years? This is an unofficial holiday of mathematicians dedicated to this interesting number, with which many formulas, mathematical and physical axioms are currently associated. This holiday was invented by the American Larry Shaw, who noticed that on this day (3.14 in the US date recording system) the famous scientist Einstein was born.

Sources:

Archimedes

Sometimes around a convex polygon you can draw it in such a way that the vertices of all the corners lie on it. Such a circle in relation to the polygon should be called circumscribed. Her center does not have to be inside the perimeter of the inscribed figure, but using the properties of the described circle, finding this point is usually not very difficult.

You will need

Ruler, pencil, protractor or square, compass.

Instructions

If the polygon around which you need to describe a circle is drawn on paper, to find center and a circle is enough with a ruler, pencil and protractor or square. Measure the length of any side of the figure, determine its middle and place an auxiliary point in this place in the drawing. Using a square or protractor, draw a segment inside the polygon perpendicular to this side until it intersects with opposite side.

Do the same operation with any other side of the polygon. The intersection of the two constructed segments will be the desired point. This follows from the main property of the described circle- her center in a convex polygon with any sides always lies at the point of intersection of the perpendicular bisectors drawn to these.

For regular polygons center and inscribed circle could be much simpler. For example, if this is a square, then draw two diagonals - their intersection will be center ohm inscribed circle. In a polygon with any even number of sides, it is enough to connect two pairs of opposite angles with auxiliary ones - center described circle must coincide with the point of their intersection. IN right triangle to solve the problem, simply determine the middle of the long side figures - hypotenuses.

If it is not known from the conditions whether, in principle, a circumscribed circle for a given polygon is possible, after determining the expected point center and using any of the described methods you can find out. Set aside the distance between the found point and any of the points on the compass, set it to the expected center circle and draw a circle - each vertex should lie on this circle. If this is not the case, then one of the properties does not hold and describe a circle around a given polygon.

Determining the diameter can be useful not only for solving geometric problems, but also help in practice. For example, knowing the diameter of the neck of a jar, you definitely won’t go wrong in choosing a lid for it. The same statement is true for larger circles.

Instructions

So, enter the notation of quantities. Let d be the diameter of the well, L the circumference, n the Pi number, the value of which is approximately 3.14, R the radius of the circle. The circumference (L) is known. Let's assume that it is 628 centimeters.

Next, to find the diameter (d), use the formula for the circumference: L = 2пR, where R is an unknown quantity, L = 628 cm, and n = 3.14. Now use the rule for finding an unknown factor: “To find a factor, you need to divide the product by a known factor.” It turns out: R=L/2p. Substitute the values into the formula: R=628/2x3.14. It turns out: R=628/6.28, R=100 cm.

Once the radius of the circle has been found (R=100 cm), use the following formula: the diameter of the circle (d) is equal to two radii of the circle (2R). It turns out: d=2R.

Now, to find the diameter, substitute d=2R values into the formula and calculate the result. Since the radius (R) is known, it turns out: d=2x100, d=200 cm.

Sources:

How to determine the diameter using the circumference of a circle

Circumference and diameter are interrelated geometric quantities. This means that the first of them can be translated into the second without any additional data. The mathematical constant through which they are related to each other is the number π.

Instructions

If the circle is represented as an image on paper and its diameter needs to be determined approximately, measure it directly. If its center is shown in the drawing, draw a line through it. If the center is not shown, find it using a compass. To do this, use a square with angles of 90 and . Attach it at a 90-degree angle to the circle so that both legs touch it, and trace it. Then applying to the resulting right angle Draw a 45 degree square angle. It will pass through the center of the circle. Then, in the same way, draw a second right angle and its bisector in another place on the circle. They will intersect in the center. This will allow you to measure the diameter.

To measure the diameter, it is preferable to use a ruler made from the thinnest possible sheet material, or a tailor's meter. If you only have a thick ruler, measure the diameter of the circle using a compass, and then, without changing its solution, transfer it to graph paper.

Also, if there are no numerical data in the conditions of the problem and if there is only a drawing, you can measure the circumference using a curvimeter, and then calculate the diameter. To use a curvimeter, first rotate its wheel to set the arrow exactly to the zero division. Then mark a point on the circle and press the curvimeter to the sheet so that the stroke above the wheel points to this point. Move the wheel along the circle line until the stroke is again above that point. Read the testimony. They will be in, bounded by a broken line. If we inscribe a regular n-gon with side b into 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=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=n*Sin (π/n) as the number of sides of an inscribed polygon tends to infinity tends to the number π, a constant value called “pi” and expressed as an infinite decimal fraction. For calculations without application computer technology the value π=3.14 is accepted. The circumference of a circle and its diameter are related by the formula: L= πD. To calculate the diameter

Circumference measurement

Scientists involved in geological research have known for a long time that our planet is spherical. That is why the first measurements of the circumference earth's surface touched the longest parallel of the Earth - the equator. This value, scientists believed, can be considered correct for any other measurement method. For example, it was believed that if you measure the circumference of the planet using the longest meridian, the resulting figure will be exactly the same.

This opinion existed until the 18th century. However, scientists from the leading scientific institution of the time - the French Academy - were of the opinion that this hypothesis was incorrect, and the shape that the planet had was not entirely correct. Therefore, in their opinion, the circumference of the longest meridian and the longest parallel will differ.

As proof, two scientific expeditions were undertaken in 1735 and 1736, which proved the truth of this assumption. Subsequently, the magnitude of the difference between these two was established - it amounted to 21.4 kilometers.

Circumference

Currently, the circumference of the planet Earth has been repeatedly measured, not by extrapolating the length of a particular segment of the earth's surface to its full size, as was done before, but using modern high-precision technologies. Thanks to this, it was possible to establish the exact circumference of the longest meridian and the longest parallel, as well as clarify the magnitude of the difference between these parameters.

So, today in the scientific community, as the official value of the circumference of the planet Earth along the equator, that is, the longest parallel, it is customary to give a figure of 40075.70 kilometers. Moreover, a similar parameter measured along the longest meridian, that is, the circumference passing through the earth’s poles, is 40,008.55 kilometers.

Thus, the difference between the circumferences is 67.15 kilometers, and the equator is the longest circumference of our planet. In addition, the difference means that one degree of the geographic meridian is slightly shorter than one degree of the geographic parallel.

Instructions

First you need the initial data for the task. The fact is that its condition cannot explicitly say what the radius is circle. Instead, the problem may give the length of the diameter circle. Diameter circle- a segment that connects two opposite points circle, passing through its center. Having analyzed the definitions circle, we can say that the length of the diameter is twice the length of the radius.

Now we can accept the radius circle equal to R. Then for the length circle you need to use the formula:
L = 2πR = πD, where L is the length circle, D - diameter circle, 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 at the points of contact with the sides of the polygon it will divide them in half. To find out the radius of the 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 using the following formula:
R = a*b*c/4S, where a, b, c are the sides of a given triangle, S is the area of the triangle around which the circle is circumscribed.
If you want to describe a circle around a quadrilateral, this can be done if two conditions are met:
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. Modern stencils include circles of different diameters. These stencils can be purchased at any office supply store.

Sources:

How to find the circumference of a circle?

A circle is a closed curved line, all points of which are at equal distances 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.

Instructions

If a straight line is drawn through the center of a circle, then its segment between two points of intersection of this line with the circle is called the diameter of the given circle. Half the diameter, from the center to the point where the diameter intersects the circle is the radius
circles. If a 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 suggests that a larger diameter outlines a larger circle bounded by a circle with a larger length. Therefore, between the diameter of a circle and its length there is a direct relationship proportional dependence.

In its physical meaning, the “circumference length” parameter corresponds to a bounded by a broken line. If we inscribe a regular n-gon with side b into 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=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=n*Sin (π/n) as the number of sides of an inscribed polygon tends to infinity tends to the number π, a constant value called “pi” 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.

Very often when deciding school assignments in physics, the question arises - how to find the circumference of a circle, knowing the diameter? In fact, there are no difficulties in solving this problem; you just need to clearly imagine what formulas,concepts and definitions are required for this.

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Basic concepts and definitions

Radius is the line connecting the center of the circle and its arbitrary point. It is denoted by the Latin letter r.
A chord is a line connecting two arbitrary points lying on a circle.
Diameter is the line connecting two points of a circle and passing through its center. It is denoted by the Latin letter d.
is a line consisting of all points located at equal distances from one selected point, called its center. We will denote its length by the Latin letter l.

The area of a circle is the entire territory enclosed within a circle. It is measured in square units and is denoted by the Latin letter s.

Using our definitions, we come to the conclusion that the diameter of a circle is equal to its largest chord.

Attention! From the definition of what the radius of a circle is, you can find out what the diameter of a circle is. These are two radii laid out in opposite directions!

Diameter of a circle.

Finding the circumference and area of a circle

If we are given the radius of a circle, then the diameter of the circle is described by the formula d = 2*r. Thus, to answer the question of how to find the diameter of a circle, knowing its radius, the last one is enough multiply by two.

The formula for the circumference of a circle, expressed in terms of its radius, has the form l = 2*P*r.

Attention! The Latin letter P (Pi) denotes the ratio of the circumference of a circle to its diameter, and this is a non-periodic decimal fraction. In school mathematics, it is considered a previously known tabular value equal to 3.14!

Now let's rewrite the previous formula to find the circumference of a circle through its diameter, remembering what its difference is in relation to the radius. It will turn out: l = 2*P*r = 2*r*P = P*d.

From the mathematics course we know that the formula describing the area of a circle has the form: s = П*r^2.

Now let's rewrite the previous formula to find the area of a circle through its diameter. We get,

s = П*r^2 = П*d^2/4.

One of the most difficult tasks in this topic is determining the area of a circle through the circumference and vice versa. Let's take advantage of the fact that s = П*r^2 and l = 2*П*r. From here we get r = l/(2*П). Let's substitute the resulting expression for the radius into the formula for the area, we get: s = l^2/(4P). In a completely similar way, the circumference is determined through the area of the circle.

Determining radius length and diameter

Important! First of all, let's learn how to measure the diameter. It's very simple - draw any radius, extend it in the opposite direction until it intersects with the arc. We measure the resulting distance with a compass and use any metric tool to find out what we are looking for!

Let us answer the question of how to find out the diameter of a circle, knowing its length. To do this, we express it from the formula l = П*d. We get d = l/P.

We already know how to find its diameter from the circumference of a circle, and we can also find its radius in the same way.

l = 2*P*r, hence r = l/2*P. In general, to find out the radius, it must be expressed in terms of the diameter and vice versa.

Suppose now you need to determine the diameter, knowing the area of the circle. We use the fact that s = П*d^2/4. Let us express d from here. It will work out d^2 = 4*s/P. To determine the diameter itself, you will need to extract square root of the right side. It turns out d = 2*sqrt(s/P).

Solving typical tasks

Let's find out how to find the diameter if the circumference is given. Let it be equal to 778.72 kilometers. Required to find d. d = 778.72/3.14 = 248 kilometers. Let's remember what a diameter is and immediately determine the radius; to do this, we divide the value d determined above in half. It will work out r = 248/2 = 124 kilometer
Let's consider how to find the length of a given circle, knowing its radius. Let r have a value of 8 dm 7 cm. Let's convert all this into centimeters, then r will be equal to 87 centimeters. Let's use the formula to find the unknown length of a circle. Then our desired value will be equal to l = 2*3.14*87 = 546.36 cm. Let's convert our obtained value into integer numbers of metric quantities l = 546.36 cm = 5 m 4 dm 6 cm 3.6 mm.
Let us need to determine the area of a given circle using the formula through its known diameter. Let d = 815 meters. Let's remember the formula for finding the area of a circle. Let's substitute the values given to us here, we get s = 3.14*815^2/4 = 521416.625 sq. m.
Now we will learn how to find the area of a circle, knowing the length of its radius. Let the radius be 38 cm. We use the formula known to us. Let us substitute here the value given to us by condition. You get the following: s = 3.14*38^2 = 4534.16 sq. cm.
The last task is to determine the area of a circle by known length circles. Let l = 47 meters. s = 47^2/(4P) = 2209/12.56 = 175.87 sq. m.

Circumference

And how is it different from a 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 indicating the boundaries of the shape 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 indicating the circle, mark two points so that they are mirror images of each other. Connect them with a line. The segment will definitely pass through the point in the center of the circle. This segment connecting opposite parts of a circle is called a diameter in geometry.

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

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

The distance from the central 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 the given figure is 14 cm.

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

This expression is also applicable as a formula for the diameter of a circle. Substituting the quantities known 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 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, apply following formula finding the diameter: D = (6 V / Pi) * 1 / 3.

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

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

We double the result obtained, taking into account that D = 2 * R.

Often you have to find the diameter of a circle in everyday life. For example, when determining what is equivalent to its diameter. To do this, you need to wrap the finger of the potential owner of the ring with thread. Mark the points of contact between the two ends. Measure the length from point to point with a ruler. We multiply the resulting value by 3.14, following the formula for determining the diameter with a known length. So, the statement that knowledge of geometry and algebra is not useful in life is not always true. And this is a serious reason for taking school subjects more responsibly.

First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on 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 limits it (circle(r)), and an innumerable 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 segment that connects two points on a circle is its 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 is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.

Central angle An angle that lies between two radii is called.

Arc length can be found using the formula:

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

The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.

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 a 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, it is called secant.

If you draw the radius to the tangent point, 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 tangent segments 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 let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external 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 a diameter, inscribed angle, right angle.

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

Inscribed angles that subtend the same arc are identical.

Inscribed angles resting on one chord are identical or their sum is equal to 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 a circle and located between two chords is identical to half the sum angular values arcs of a circle that are contained within a given and vertical angle.

\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 values of the arcs of the circle that are contained 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 a polygon.

At the point where the bisectors of the corners of a 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 semi-perimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is equal to:

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 fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.

AB + DC = AD + BC

It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors of the internal 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)

Circumcircle

If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.

At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.

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

Eat next condition: a circle can be described 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)

Around any triangle you can describe a circle, 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.