Determine the circumference of the circle of the radius of chord diameter. What is a circle as a geometric shape: basic properties and characteristics

We see that such a circle and a circle. The formula of the area of \u200b\u200bthe circle and the length of the circle.

We meet many items every day, in the form that form a circle or opposite the circle. Sometimes there is a question that is a circle and how it differs from the circle. Of course, we all passed geometry lessons, but sometimes it would not hurt to refresh the knowledge of very simple explanations.

What is the circumference of the circle and the area of \u200b\u200bthe circle: definition

So, the circle is a closed line curve, which limits or on the contrary, forms a circle. A mandatory circumference condition - she has a center and all the points are equidistant from it. Simply put, the circle is a gymnastic hoop (or as they are often called Hula-HUP) on a flat surface.

The circumference of the circumference is the total length of the very curve that forms a circle. As is known, regardless of the size of the circle, the ratio of its diameter and the length is equal to the number π \u003d 3,141592653589793238462643.

From this it follows that π \u003d L / D, where L is the circumference length, and D is the diameter of the circle.

If the diameter is known to you, then the length can be found on a simple formula: L \u003d π * D

In case the radius is known: L \u003d 2 ™

We figured out what a circle is and can proceed to the definition of the circle.

The circle is a geometric shape that is surrounded by a circle. Or, the circle is a figure, the turn of which consists of a large number of points equidistant from the center of the figure. The whole area, which is inside the circle, including its center, is called a circle.

It is worth noting that the circumference and the circle, which is in it the values \u200b\u200bof the radius and the diameter of the same one. And the diameter in turn is two times more than the radius.

The circle has an area on the plane that can be found using a simple formula:

Where S is the area of \u200b\u200bthe circle, and R is the radius of this circle.

What the circle is different from the circle: explanation

The main difference between the circle and the circle is that the circle is a geometric figure, and the circle is a closed curve. Also pay attention to the differences between the circle and the circle:

• Circle is a closed line, and the circle is an area inside this circle;
• Circle is a curve line on the plane, and the circle is space closed in a ring of a circle;
• Similarity between the circumference and the circle: radius and diameter;
• In the circle and circle, a single center;
• If the space is shaded inside the circle, it turns into a circle;
• The circle has a length, but there is no circle, and on the contrary, the circle has an area that does not have a circle.

Circle and circle: examples, photo

For clarity, we propose to consider the photo on which the circle is shown on the left, and the right circumference.

Formula of the length of the circle and square area: comparison

The formula of the circumference of the circumference L \u003d 2 πr

Formula Square S \u003d πr²

Please note that in both formulas there is a radius and number π. These formulas are recommended to learn by heart, as they are the simplest and will be useful in everyday life and at work.

Circle area in the length of the circle: Formula

S \u003d π (L / 2π) \u003d L² / 4π, where S is the area of \u200b\u200bthe circle, L is the circumference length.

Video: What is a circle, circle and radius

Circle - This is a figure that consists of all points of the plane equidistant from this point.

Basic concepts:

Center Circle - This is a point equid to the circumference points.

Radius - This is the distance from the circumference points to its center (equal to half of the diameter, Fig. 1).

Diameter - This is a chord passing through the center of the circle (Fig. 1).

Chord - This is a segment connecting two circumference points (Fig. 1).

Tangent - This is a straight line, having only one common point with a circle. It passes through the circumference point perpendicular to the diameter carried out at this point (Fig. 1).

Secant - It is straight, passing through two different circumference points (Fig. 1).

Single circle - This is a circle, the radius of which is equal to one.

Arc area - This is a part of the circumference, separated by two inconsistent points of the circle.

1 radian - It is an angle formed by an arc circumference, equal to the length of the radius (Fig. 4).
1 radian \u003d 180˚: π ≈ 57.3˚

Central corner - This is an angle with a vertex in the center of the circumference. It is equal to the degree of arc, which relies (Fig. 2).

Inserted corner - This is the corner, the top of which lies on the circle, and the sides crosses this circle. Equal to half a degree of arc, which relies (Fig. 3).

Two circles having a common center called concentric.

Two circles intersecting at right angles are called orthogonal.

Circle Length and Circle Area:

Designations:
Circle Length - C
Diameter Length - D
Radius length - R

Valueπ :
The ratio of the circumference of the circumference to the length of its diameter is indicated by the Greek letter π (PI).

22
π = -
7

The formula of the circumference of the circumference:

C \u003d πd, or c \u003d 2πr

Circle area formulas:

C · R.
S \u003d -
2

π · d 2
S \u003d ---
4

The area of \u200b\u200bthe circular sector and the circular segment.

Circular equation in Cartesian coordinatesx., y. c center at point (a; b.):

(x -a.) 2 + (y - B.) 2 = R. 2

The circle described near the triangle (Fig. 4).

The circle inscribed in the triangle (Fig. 5).

Corners inscribed in a circle (Fig. 3).

Angle whose vertex lies on the circle, and the sides crosses this circle, called inserted into the circle.

Basic concepts:

The angle divides the plane into two parts. Each of these parts is called flat corner.

Flat corners with shared parties are called additional.

Flat corner with a vertex in the center of the circle is called central angle(Fig.2)

Proportionality of chord segments and sequencing circumference.

Private cases and formulas:

1) from the point C, which is outside the circle, we carry out a tangent of the circle and denote the point of their contact of the letter D.

Then, from the same point C, we carry out the securing and intersection points of the unit and the circumference with letters A and B (Fig. 8).

In this case:

CD 2 \u003d.AC · BC.

2) Cut the AB diameter in the circle. Then, from the point C, which is in the circle, we carry out perpendicular to this diameter and denote the resulting CD segment (Fig. 9).

In this case:

CD 2 \u003d.Ad · BD.

Circle - geometric shape consisting of all points of the plane located at a given distance from this point.

This point (O) is called center Circle.
Radius of circle - This is a segment connecting the center with any point of the circle. All radii have the same length (by definition).
Chord - Cut connecting two circumference points. Chord, passing through the center of the circle, is called diameter. The center of the circle is the middle of any diameter.
Any two circumference points divide it into two parts. Each of these parts is called arc circle. Arc is called semi-rapidIf the segment connecting its ends is a diameter.
The length of a single semicondancy is denoted by π .
The sum of the degree of two arcs of circle with general ends is equal 360º.
Part of the plane limited by a circle called around.
Circular sector - Part of a circle bounded by arc and two radii connecting the ends of the arc with the center of the circle. Arc that limits the sector is called arc sector.
Two circles having a common center called concentric.
Two circles intersecting at right angles are called orthogonal.

Mutual arrangement of direct and circumference

1. If the distance from the center of the circumference to the straight line is less than the circle radius ( d), then direct and circle have two common points. In this case, the direct is called sale in relation to the circumference.
2. If the distance from the center of the circle to the straight is equal to the radius of the circumference, then the direct and circle have only one common point. Such a direct is called tangent to circumference, and their common point is called point of touch of direct and circumference.
3. If the distance from the center of the circle to the straight line is more than the circle radius, then direct and circle do not have common points
.

Central and inscribed angles

Central corner - This is an angle with a vertex in the center of the circumference.
Inserted corner - The angle whose vertex lies on the circle, and the sides crosses the circle.

Theorem on the inscribed coal

The inscribed angle is measured half an arc to which it relies.

• Corollary 1.
  Inserted angles, relying on the same arc equal.


• Corollary 2.
  The inscribed angle based on the semicircle is straight.

Theorem on the product of segments of intersecting chords.

If two chords of the circumference intersect, then the product of segments of one chord is equal to the product of segments of another chord.

Basic formulas

• Circumference:

• Circle arc length:

• Diameter:

• Circle arc length:

• Area of \u200b\u200ba circle:

• Circular Square:

Equation of the circle

• In the rectangular coordinate system, the equation of the circle of the radius r. with center at point C. (x o; y o) has the form:

• The equation of the circle of the radius R with the center at the beginning of the coordinate has the form:

The circle is called a closed line curve on the plane, all points of which are at the same distance from one point; This point is called the center of the circle.

Part of the plane limited by the circle is called a circle.

A straight line connecting the circumference point with its center is called a radius (Fig. 84).

Since all the circumference points are from the center at the same distance, all the radii of the same circumference are equal to each other. Radius is usually indicated by the letter R. or r..

The point taken inside the circle is located from its center at a distance of a smaller radius. This is easy to make sure if it is possible to carry out a radius (Fig. 85) through this point.

The point taken out of the circle is located from its center at a distance, larger radius. This is easy to make sure if you connect this point with the center of the circle (Fig. 85).

A straight line connecting two circumference points is called chord.

Chord passing through the center is called a diameter (Fig. 84). The diameter is usually indicated by the letter D. The diameter is equal to two radius:

Since all the radii of the same circle is equal to each other, then all the diameters of this circle are equal to each other.

Theorem. Chord, not passing through the center of the circle, less diameter spent in the same circle.

In fact, if you do any chord, for example, AB, and connect its ends with the center of O (Fig. 86), we will see that the chord of AB is less than the broken line of AO + OV, that is, AB R, and since 2. r. \u003d D, then av

If the circle is overloaded in diameter (Fig. 87), then both parts of the circle and circle are monitored. The diameter divides the circle and circle into two equal parts.

Two circles (two circles) are called equal, if they can be supplied to each other so that they are combined.

Therefore, two circles (two circles) with equal radii are equal.

2. Arc area.

Part of the circle is called arc.

The word "arc" is sometimes replaced by the sign \\ (\\ breve () \\). The arc is denoted by two or three letters, of which two are put at the ends of the arc, and the third is some point of the arc. The drawing 88 indicates two arcs: \\ (\\ Breve (ACB) \\) and \\ (\\ Breve (ADB) \\).

In the event that an arc is less than a semicircle, it is usually indicated by two letters. So, an ADV arc can be denoted \\ (\\ Breve (AB) \\) (Fig. 88). About the chord, which connects the ends of the arc, they say that she tightens the arc.

If you move the arc of the AC (Fig. 89, a) so that it slides the submitted circumference, and if it coincides with the MN arc, then \\ (\\ Breve (AC) \\) \u003d \\ (\\ Breve (Nm) \\).

In the drawing 89, the arc of the AC and AV are not equal to each other. Both arcs begin at point A, but one arc \\ (\\ breve (AB) \\) is only part of another arc \\ (\\ breve (AC) \\).

Therefore, \\ (\\ Breve (AC) \\)\u003e \\ (\\ BREVE (AB) \\); \\ (\\ BREVE (AB) \\)

Building a circle for three points

A task. After three points that are not lying on one straight line, conduct a circle.

Let us give three points A, B and C, not lying on one straight line (Damn.311).

Connect these points with segments AB and Sun. To find points equidistant from points A and in the separation of the segment of AB in half and through the middle (point M), we will spend direct perpendicular to av. Each point of this perpendicular is equally removed from the points A and V.

To find points that are equidistant from the points in and C, we divide the segment of the Sun in half and through its middle (point N) will spend direct, perpendicular to sun. Each point of this perpendicular is equally removed from the points B and C.

The point of intersection of these perpendicular will be at the same distance from these points A, B and C (AO \u003d CO). If we, accepting the point about the center of the circle, with a radius equal to JSC, carry out a circle, then it will pass through all the data of points A, B and C.

The point O is the only point that can serve as the center of the circle passing through three points A, B and C, which are not lying on one straight line, as two perpendicular to the segments of AB and Sun can cross only at one point. So the task has a single solution.

Note. If three points A, B and C will lie on one straight line, the task will not have solutions, since perpendicular to segments of AB and Sun will be parallel and there will be no points equally removed from points A, B, C, i.e. . The points that could serve as the center of the desired circumference.

If you connect the segment of the point A and C and the middle of this segment (point K) to combine with the center of the circle O, the OK will be perpendicular to the AU (Damn 311), since the AOC is a median, therefore, OKA is a median triangle.

Corollary. Three perpendicular to the sides of the triangle, conducted through their middle intersect at one point.

Circle - This is a figure that consists of all points on the plane equidistant from this point. This point is called the center of the circle.

The circle of zero radius (degenerate circle) is a point, sometimes this case is excluded from the definition.

Encyclopedic YouTube.

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Circle and properties (bezbotvy)

Inscribed and described circle - from Bezbotvy

Mathematics: Preparation for OGE and EGE. Planimetry. Circle and their properties

Mathematics 26. Circle. Circle and Circle - Shishkin School

Circle equation. Task 18 (C5). Arthur Sharifov

Subtitles

Designation

If the circle passes, for example, through points A, B, C, it is indicated by the indication of these points in parentheses: (A, B, C). Then the arc of the circle passing through the points A, B, C is denoted as an ABC arc (or an AC arc), as well as υ ABC (or υ AC).

Other definitions

• Circle diameter AB A, B. AB Viden at a right angle (determining via an angle, based on the diameter of the circle).

• Curl with chorda AB - This is a figure consisting of points A, B. and all points of the plane, of which the segment AB visible at a permanent corner on one side equal inscribed angle of arc AB, and under a different permanent angle on the other hand, equal to 180 degrees minus inscribed arc angle ABspecified above (definition through the inscribed angle).
• Figure consisting of such points X, (\\ DisplayStyle X,) that the ratio of lengths of segments AX. and BX. constantly: A x b x \u003d c ≠ 1, (\\ displaystyle (\\ FRAC (AX) (BX)) \u003d C \\ NEQ 1,) It is a circle (determination through the Circle of Apollonia).
• The figure consisting of all such points for each of which the sum of the squares of the distance to two points of the points is equal to a given value, the greater half of the square of the distance between these points is also a circle (determining through the Pythagora theorem for an arbitrary rectangular triangle, inscribed in a circle, with hypotenuse which is the diameter of the circle).
• M. Inside her spend any chords AB, CD, EF. and so on, then equality are true: A M ⋅ M B \u003d C M ⋅ M D \u003d E M ⋅ M F \u003d ... (\\ DisplayStyle Am \\ Cdot (MB) \u003d CM \\ CDOT (MD) \u003d EM \\ CDOT (MF) \u003d \\ DOTS). Equality will always be executed independently of the point M. and directions conducted through her chord (definition through intersecting chords).
• The circle is a closed, self-appliant figure with the following property. If through an arbitrary point M. beyond it to spend two tangents to their touchpoints with a circle, for example, A. and B., then their lengths will always be equal: M a \u003d m b (\\ DisplayStyle Ma \u003d MB). Equality will always be executed regardless of the point selection M. (Definition through equal tangents).
• The circle is a closed, self-appliant figure with the following property. The ratio of the length of any of her chords to the sinus of anyone inscribed cornerBased on this chord, there is a constant value equal to the diameter of this circle (definition through the sinus theorem).
• The circle is a special case of an ellipse, in which the distance between focus is zero (definition through degenerate ellipse).

Related definitions for one circle

• The geometric location of the plane points, the distance from which is not more than the specified nonzero, is called around .
• Radius - Not only distance, but also a segment connecting the center of the circle with one of its points. Radius is always equal to half diatera Circle.
• The radius is always perpendicular to a tangent direct, conducted to the circle in its common point with a circle. That is, the radius is both normal to the circumference.
• Circle called single if its radius is equal to one. Single circle It is one of the main objects of trigonometry.
• Cut connecting two circumference points is called it chordoy. Chord, passing through the center of the circle, is called diameter.
• Any two do not match the circumference points are divided into two parts. Each of these parts is called arc circle . Arc is called semi-rapidIf the segment connecting its ends is a diameter.
• The length of a single semicondancy is denoted by.

• Direct, having a circle exactly one common point called tangent To the circumference, and their common point is called a direct and circumference point.
• Tangent to the circle is always perpendicular to its radius (and diameter) carried out at the point of touch, which is normalconducted at this point.
• Direct, passing through two different circumference points, is called sale.

Definition of triangles for one circle

• ABC triangle is called inserted into the circle (A, B, C) if all three of its vertices a, b and c lie on this circle. In this case, the circle is called described circle ABC triangle (see described circle).
• Tangent To the circumference, carried out through any vertex of an anti-parallel dealer inscribed in her, the side of the triangle opposite to this vertex.
• ABC triangle is called described near the circle (A ", B", C ") if all three of it are AB, BC and CAs relate to this circle at some points, respectively, C", A "and B". In this case, the circle is called in addition to the circle ABC triangle (see inscribed circle).

Corner definitions for one circle

• The angle formed by the arc circle equal to the length of the radius is taken for 1 radian.

• Central Corner - angle with a top in the center of the circle. The central angle is radiable / degree of arc, which relies (see Fig.).
• InscribedThe angle is the angle, the top of which lies on the circle, and the sides crosses this circle. Inserted corner It is equal to half a degree of arc, which relies (see Fig.).
• Outdoor for Inscribedangle - an angle formed by one side and continuation of the other side inscribed angle (see fig. Corner θ Brown color). Outdoor For inscribed on the other side, the angle of the circle has the same magnitude θ .
• The angle between the circle and direct - The angle between the straight and tangent to the circumference at the point of intersection of the direct and circle. Both corners between intersecting circles and direct are equal.
• Angle based on circle diameter - The angle inscribed in this circle, the sides of which contain the end of the diameter. He is always straight.

Related definitions for two circles

• Two circles having a common center called concentric.
• Two circles having only one common point are called touching Externally, if their circles do not have other common points, and internal way if their circles lie one inside the other.
• Two circles having two common points are called crossing. Their circles (asimes are limited) intersect in a region called a double circular segment.
• Angle Between two intersecting (or relating) circles called the angle between their tangent, carried out in the overall intersection point (or touch).
• Also angle Between two intersecting (or relating) circles, you can read the angle between their radii (diameters) carried out in the overall intersection point (or touch).
• Because for any circumference, its radius (or diameter) and tangential, carried out through any point of the circle, mutually perpendicular to the radius (or diameter) can be considered normal To the circle built at this point. Consequently, two types of angles defined in the two previous two points will always be equal to each other, as angles with mutually perpendicular sides.
• direct angle called orthogonal. Circle can be considered orthogonalIf they form a straight corner with each other.
• Radical axis of two circles - The geometrical point of points, the degrees of which relative to two given circles are equal. In other words, equal to the length of four tangents conducted to the two data circles from anywhere M. This geometric point of points.

Corner definitions for two circles

• The angle between two intersecting circles - angle between tangent to circles at the intersection point of these circles. Both angles between two intersecting circles are equal.
• The angle between two non-cycle circles - The angle between the two communal to two circles, formed at the point of intersection of these two tangents. The intersection point of these two tangents should lie between two circles, and not from one of them (this angle is not considered). Both vertical angle between the two non-rigging circles are equal.

Orthogonality

• Two circles intersecting at right angles are called orthogonal. Circle can be considered orthogonalIf they form a straight corner with each other.
• Two intersecting at points A and B circle with centers O and O are called orthogonalIf there are direct angles of OAO "and OBO". This condition guarantees right angle between the circles. In this case, perpendicular to the radii (normal) of the two circles, carried out to the point of their intersection. Therefore, perpendicular to the tangent two circles, carried out in the point of their intersection. The tangent circumference is perpendicular to the radius (normal), spent on the touch point. Usually the angle between curves is the angle between their tangents spent at the point of their intersection.
• Another additional condition is possible. Let two intersecting at points a and b circumferences have meaningful arcs at points C and D, that is, the AS arc is equal to the CB arc, the AD arc is equal to the DB arc. Then these circles are called orthogonalIf you are direct CAD and CBD angles.

Related definitions for three circles

• Three circles are called mutually relating (supplied) if any two of them relate (stopped) each other.
• In geometry radical Center Three circles are the point of intersection of three radical axes of couples of circles. If the radical center lies outside of all three circles, then it is the center of the only circumference ( radical circle), which crosses three circumference data orthogonal.

Lemma Archimedes

Evidence

Let be G (\\ DisplayStyle G) - Homotheusy, which translates a small circle to the greater. Then it is clear that A 1 (\\ DisplayStyle A_ (1)) He is the center of this homotheetics. Then straight B C (\\ DisplayStyle BC) will go to some straight A (\\ DisplayStyle A)concerning a large circumference, and A 2 (\\ DisplayStyle A_ (2)) Go to the point on this direct and owned large circumference. Remembering that the homothety translates the straight lines in parallel to them, we understand that A ∥ B C (\\ DisplayStyle A \\ Parallel BC). Let be G (A 2) \u003d A 3 (\\ DisplayStyle G (A_ (2)) \u003d A_ (3)) and D (\\ DisplayStyle D) - point on a straight A (\\ DisplayStyle A), such that - sharp, and E (\\ DisplayStyle E) - such a point on a straight A (\\ DisplayStyle A), what ∠ B A 3 E (\\ DisplayStyle \\ Angle BA_ (3) E) - sharp. Then because A (\\ DisplayStyle A) - tangent to great circle ∠ C A 3 D (\\ DisplayStyle \\ Angle CA_ (3) D) \u003d (\\ displayStyle \u003d) ∠ C B A 3 (\\ DisplayStyle \\ Angle CBA_ (3)) \u003d ∠ B A 3 E \u003d ∠ B C A 3 (\\ DISPLAYSTYLE \u003d \\ ANGLE BA_ (3) E \u003d \\ ANGLE BCA_ (3)). Hence △ B C A 3 (\\ DisplayStyle \\ Bigtriangleup BCA_ (3)) - equiced, and therefore ∠ B A 1 A 3 \u003d ∠ C A 1 A 3 (\\ DISPLAYSTYLE \\ ANGLE BA_ (1) A_ (3) \u003d \\ ANGLE CA_ (1) A_ (3)), i.e A 1 A 2 (\\ DISPLAYSTYLE A_ (1) A_ (2)) - bisector corner ∠ B A 1 C (\\ DisplayStyle \\ Angle BA_ (1) C).

Decartes theorem for radii four in pairwise concerning circles

Decartes Theorem " It claims that the radii of any four mutually relating circles satisfy some square equation. They are sometimes called Soddy's circles.

Properties

The equation of the circle passing through points (x 1, y 1), (x 2, y 2), (x 3, y 3), (\\ displaystyle \\ left (x_ (1), y_ (1) \\ right), \\ left (x_ (2) , y_ (2) \\ right), \\ left (x_ (3), y_ (3) \\ right),) Do not lying on one straight line (using the determinant):

In the Cartesian coordinate system, the circle is not a graph schedule, but it can be described as a combination of graphs of the two following functions:

If the center of the circle coincides with the start of coordinates, the functions take the form:

Polar coordinates

Radius circle R (\\ DisplayStyle R) with center at point (ρ 0, φ 0) (\\ displaystyle \\ left (\\ rho _ (0), \\ phi _ (0) \\ Right)).