The part of a circle bounded by two points is called. How does a circle differ from a circle: explanation

Circle is a flat closed line, all points of which are at the same distance from a certain point (point O), which is called the center of the circle.
(A circle is a geometric figure consisting of all points located at a given distance from a given point.)

Circle is a part of the plane limited by a circle. Point O is also called the center of the circle.

The distance from a point on a circle to its center, as well as the segment connecting the center of the circle to its point, is called the radius circle/circle.
See how circle and circumference are used in our life, art, design.

Chord - Greek - a string that binds something together
Diameter - "measurement through"

ROUND FORM

Angles can occur in ever-increasing quantities and, accordingly, acquire an ever-increasing turn - until they completely disappear and the plane becomes a circle.This is a very simple and at the same time very complex case, which I would like to talk about in detail. It should be noted here that both simplicity and complexity are due to the absence of angles. The circle is simple because the pressure of its boundaries, in comparison with rectangular shapes, is leveled - the differences here are not so great. It is complex because the top imperceptibly flows into the left and right, and the left and right into the bottom.

V. Kandinsky

In Ancient Greece, the circle and circumference were considered the crown of perfection. Indeed, at each point the circle is arranged in the same way, which allows it to move on its own. This property of the circle made the wheel possible, since the axle and hub of the wheel must be in contact at all times.

Many useful properties of a circle are studied at school. One of the most beautiful theorems is the following: let us draw a line through a given point intersecting a given circle, then the product of the distances from this point to the intersection points of a circle with a straight line does not depend on exactly how the straight line was drawn. This theorem is about two thousand years old.

In Fig. Figure 2 shows two circles and a chain of circles, each of which touches these two circles and two neighbors in the chain. The Swiss geometer Jacob Steiner proved the following statement about 150 years ago: if the chain is closed for a certain choice of the third circle, then it will be closed for any other choice of the third circle. It follows from this that if the chain is not closed once, then it will not be closed for any choice of the third circle. To the artist who painteddepicted chain, one would have to work hard to make it work, or turn to a mathematician to calculate the location of the first two circles, at which the chain is closed.

We mentioned the wheel first, but even before the wheel, people used round logs- rollers for transporting heavy loads.

Is it possible to use rollers of some other shape than round? Germanengineer Franz Relo discovered that rollers, the shape of which is shown in Fig., have the same property. 3. This figure is obtained by drawing arcs of circles with centers at the vertices of an equilateral triangle, connecting two other vertices. If we draw two parallel tangents to this figure, then the distance betweenthey will be equal to the length of the side of the original equilateral triangle, so such rollers are no worse than round ones. Later, other figures were invented that could serve as rollers.

Enz. "I explore the world. Mathematics", 2006

Each triangle has, and moreover, only one, nine point circle. Thisa circle passing through the following three triplets of points, the positions of which are determined for the triangle: the bases of its altitudes D1 D2 and D3, the bases of its medians D4, D5 and D6the midpoints of D7, D8 and D9 of straight segments from the point of intersection of its heights H to its vertices.

This circle, found in the 18th century. by the great scientist L. Euler (which is why it is often also called Euler’s circle), was rediscovered in the next century by a teacher at a provincial gymnasium in Germany. This teacher's name was Karl Feuerbach (he was the brother of the famous philosopher Ludwig Feuerbach).Additionally, K. Feuerbach found that a circle of nine points has four more points that are closely related to the geometry of any given triangle. These are the points of its contact with four circles of a special type. One of these circles is inscribed, the other three are excircles. They are inscribed in the corners of the triangle and externally touch its sides. The points of tangency of these circles with the circle of nine points D10, D11, D12 and D13 are called Feuerbach points. Thus, the circle of nine points is actually the circle of thirteen points.

This circle is very easy to construct if you know its two properties. Firstly, the center of the circle of nine points lies in the middle of the segment connecting the center of the circumscribed circle of the triangle with point H - its orthocenter (the point of intersection of its altitudes). Secondly, its radius for a given triangle is equal to half the radius of the circle circumscribed around it.

Enz. reference book for young mathematicians, 1989

We see circle shapes and circles everywhere: this is the wheel of a car, the horizon line, and the disk of the Moon. Mathematicians began to study geometric figures - a circle on a plane - a very long time ago.

A circle with a center and radius is a set of points on a plane located at a distance not greater than . A circle is bounded by a circle consisting of points located exactly at a distance from the center. The segments connecting the center with the points of the circle have a length and are also called radii (of a circle, circle). The parts of the circle into which it is divided by two radii are called circular sectors (Fig. 1). A chord - a segment connecting two points on a circle - divides the circle into two segments, and the circle into two arcs (Fig. 2). A perpendicular drawn from the center to the chord divides it and the arcs subtended by it in half. The chord is longer, the closer it is located to the center; the longest chords - the chords passing through the center - are called diameters (of a circle, circle).

If a straight line is removed from the center of the circle by a distance , then at it does not intersect with the circle, at it intersects with the circle along a chord and is called a secant, at at it has a single common point with the circle and the circle and is called a tangent. A tangent is characterized by the fact that it is perpendicular to the radius drawn to the point of tangency. Two tangents can be drawn to a circle from a point outside it, and their segments from a given point to the points of tangency are equal.

Arcs of a circle, like angles, can be measured in degrees and fractions. Part of the entire circle is taken as a degree. The central angle (Fig. 3) is measured in the same number of degrees as the arc on which it rests; an inscribed angle is measured by half an arc. If the vertex of an angle lies inside the circle, then this angle in degrees is equal to half the sum of the arcs and (Fig. 4, a). An angle with a vertex outside the circle (Fig. 4,b), cutting out arcs and on the circle, is measured by the half-difference of arcs and. Finally, the angle between the tangent and the chord is equal to half the arc of a circle enclosed between them (Fig. 4, c).

A circle and a circle have an infinite number of axes of symmetry.

From the theorems on the measurement of angles and the similarity of triangles follow two theorems on proportional segments in a circle. The chord theorem says that if a point lies inside a circle, then the product of the lengths of the segments of chords passing through it is constant. In Fig. 5,a. The theorem about secant and tangent (meaning the lengths of segments of parts of these straight lines) states that if a point lies outside the circle, then the product of the secant and its external part is also unchanged and equal to the square of the tangent (Fig. 5,b).

Even in ancient times, they tried to solve problems related to the circle - to measure the length of a circle or its arc, the area of a circle or sector, segment. The first of them has a purely “practical” solution: you can lay a thread along a circle, and then unroll it and attach it to a ruler, or mark a point on the circle and “roll” it along the ruler (you can, on the contrary, “roll” a circle with a ruler). One way or another, measurements showed that the ratio of the circumference to its diameter is the same for all circles. This ratio is usually denoted by a Greek letter (“pi” is the initial letter of the Greek word perimetron, which means “circle”).

However, the ancient Greek mathematicians were not satisfied with such an empirical, experimental approach to determining the circumference of a circle: a circle is a line, i.e., according to Euclid, “length without width,” and such threads do not exist. If we roll a circle along a ruler, then the question arises: why do we get the circumference and not some other value? In addition, this approach did not allow us to determine the area of the circle.

The solution was found as follows: if we consider regular -gons inscribed in a circle, then as , tending to infinity, in the limit they tend to . Therefore, it is natural to introduce the following, already strict, definitions: the length of a circle is the limit of the sequence of perimeters of regular triangles inscribed in a circle, and the area of a circle is the limit of the sequence of their areas. This approach is also accepted in modern mathematics, and in relation not only to the circle and circle, but also to other curved areas or areas limited by curvilinear contours: instead of regular polygons, sequences of broken lines with vertices on curves or contours of areas are considered, and the limit is taken when the length tends to the greatest links of the broken line to zero.

The length of a circular arc is determined in a similar way: the arc is divided into equal parts, the division points are connected by a broken line, and the length of the arc is assumed to be equal to the limit of the perimeters of such broken lines as , tending to infinity. (Like the ancient Greeks, we do not clarify the concept of limit itself - it no longer refers to geometry and was quite strictly introduced only in the 19th century.)

From the definition of the number itself, the formula for the circumference follows:

For the length of an arc, we can write a similar formula: since for two arcs and with a common central angle, the proportion follows from considerations of similarity, and from it the proportion, after passing to the limit we obtain the independence (of the radius of the arc) of the ratio. This ratio is determined only by the central angle and is called the radian measure of this angle and all corresponding arcs with center at. This gives the formula for the arc length:

where is the radian measure of the arc.

The written formulas for and are just rewritten definitions or notations, but with their help we obtain formulas for the areas of a circle and a sector that are far from just notations:

To derive the first formula, it is enough to go to the limit in the formula for the area of a regular triangle inscribed in a circle:

By definition, the left side tends to the area of the circle, and the right side tends to the number

and , bases of its medians and , midpoints and line segments from the point of intersection of its heights to its vertices.

This circle, found in the 18th century. by the great scientist L. Euler (which is why it is often also called Euler’s circle), was rediscovered in the next century by a teacher at a provincial gymnasium in Germany. This teacher's name was Karl Feuerbach (he was the brother of the famous philosopher Ludwig Feuerbach). Additionally, K. Feuerbach found that a circle of nine points has four more points that are closely related to the geometry of any given triangle. These are the points of its contact with four circles of a special type (Fig. 2). One of these circles is inscribed, the other three are excircles. They are inscribed in the corners of the triangle and externally touch its sides. The points of contact of these circles with a circle of nine points are called Feuerbach points. Thus, the circle of nine points is actually the circle of thirteen points.

This circle is very easy to construct if you know its two properties. Firstly, the center of the circle of nine points lies in the middle of the segment connecting the center of the circle circumscribed about the triangle with a point - its orthocenter (the point of intersection of its altitudes). Secondly, its radius for a given triangle is equal to half the radius of the circle circumscribed around it.

AND circle- geometric shapes interconnected. there is a boundary broken line (curve) circle,

Definition. A circle is a closed curve, each point of which is equidistant from a point called the center of the circle.

To construct a circle, an arbitrary point O is selected, taken as the center of the circle, and a closed line is drawn using a compass.

If point O of the center of the circle is connected to arbitrary points on the circle, then all the resulting segments will be equal to each other, and such segments are called radii, abbreviated by the Latin small or capital letter “er” ( r or R). You can draw as many radii in a circle as there are points in the length of the circle.

A segment connecting two points on a circle and passing through its center is called a diameter. Diameter consists of two radii, lying on the same straight line. Diameter is indicated by the Latin small or capital letter “de” ( d or D).

Rule. Diameter a circle is equal to two of its radii.

d = 2r
D=2R

The circumference of a circle is calculated by the formula and depends on the radius (diameter) of the circle. The formula contains the number ¶, which shows how many times the circumference is greater than its diameter. The number ¶ has an infinite number of decimal places. For calculations, ¶ = 3.14 was taken.

The circumference of a circle is denoted by the Latin capital letter “tse” ( C). The circumference of a circle is proportional to its diameter. Formulas for calculating the circumference of a circle based on its radius and diameter:

C = ¶d
C = 2¶r

Examples
Given: d = 100 cm.
Circumference: C=3.14*100cm=314cm
Given: d = 25 mm.
Circumference: C = 2 * 3.14 * 25 = 157 mm

Circular secant and circular arc

Every secant (straight line) intersects a circle at two points and divides it into two arcs. The size of the arc of a circle depends on the distance between the center and the secant and is measured along a closed curve from the first point of intersection of the secant with the circle to the second.

Arcs circles are divided secant into a major and a minor if the secant does not coincide with the diameter, and into two equal arcs if the secant passes along the diameter of the circle.

If a secant passes through the center of a circle, then its segment located between the points of intersection with the circle is the diameter of the circle, or the largest chord of the circle.

The further the secant is located from the center of the circle, the smaller the degree measure of the smaller arc of the circle and the larger the larger arc of the circle, and the segment of the secant, called chord, decreases as the secant moves away from the center of the circle.

Definition. A circle is a part of a plane lying inside a circle.

The center, radius, and diameter of a circle are simultaneously the center, radius, and diameter of the corresponding circle.

Since a circle is part of a plane, one of its parameters is area.

Rule. Area of a circle ( S) is equal to the product of the square of the radius ( r 2) to the number ¶.

Examples
Given: r = 100 cm
Area of a circle:
S = 3.14 * 100 cm * 100 cm = 31,400 cm 2 ≈ 3 m 2
Given: d = 50 mm
Area of a circle:
S = ¼ * 3.14 * 50 mm * 50 mm = 1,963 mm 2 ≈ 20 cm 2

If you draw two radii in a circle to different points on the circle, then two parts of the circle are formed, which are called sectors. If you draw a chord in a circle, then the part of the plane between the arc and the chord is called circle segment.

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

This point (O) is called center of the circle.
Circle radius- this is a segment connecting the center with any point on the circle. All radii have the same length (by definition).
Chord- a segment connecting two points on a circle. A chord passing through the center of a circle is called diameter. The center of a circle is the midpoint of any diameter.
Any two points on a circle divide it into two parts. Each of these parts is called arc of a circle. The arc is called semicircle, if the segment connecting its ends is a diameter.
The length of a unit semicircle is denoted by π .
The sum of the degree measures of two arcs of a circle with common ends is equal to 360º.
The part of the plane bounded by a circle is called all around.
Circular sector- a part of a circle bounded by an arc and two radii connecting the ends of the arc to the center of the circle. The arc that limits the sector is called arc of the sector.
Two circles having a common center are called concentric.
Two circles intersecting at right angles are called orthogonal.

The relative position of a straight line and a circle

If the distance from the center of the circle to the straight line is less than the radius of the circle ( d), then the straight line and the circle have two common points. In this case the line is called secant in relation to the circle.
If the distance from the center of the circle to the straight line is equal to the radius of the circle, then the straight line and the circle have only one common point. This line is called tangent to the circle, and their common point is called point of tangency between a line and a circle.
If the distance from the center of the circle to the straight line is greater than the radius of the circle, then the straight line and the circle have no common points

Central and inscribed angles

Central angle is an angle with its vertex at the center of the circle.
Inscribed angle- an angle whose vertex lies on a circle and whose sides intersect the circle.

Inscribed angle theorem

An inscribed angle is measured by the half of the arc on which it subtends.

Corollary 1.
Inscribed angles subtending the same arc are equal.

Corollary 2.
An inscribed angle subtended by a semicircle is a right angle.

Theorem on the product of segments of intersecting chords.

If two chords of a circle intersect, then the product of the segments of one chord is equal to the product of the segments of the other chord.

Basic formulas

Circumference:

C = 2∙π∙R

Circular arc length:

R = С/(2∙π) = D/2

Diameter:

D = C/π = 2∙R

Circular arc length:

l = (π∙R) / 180∙α,
Where α - degree measure of the length of a circular arc)

Area of a circle:

S = π∙R 2

Area of the circular sector:

S = ((π∙R 2) / 360)∙α

Equation of a circle

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

(x - x o) 2 + (y - y o) 2 = r 2

The equation of a circle of radius r with center at the origin has the form:

x 2 + y 2 = r 2

A circle with center $O$ and radius $R$ is the set of points in the plane removed from $O$ at a distance not greater than $R.$ A circle is bounded by a circle consisting of points removed from the center $O$ exactly at a distance $R .$ The segments connecting the center with the points of the circle have length $R$ and are also called radii (of a circle, circle). The parts of the circle into which it is divided by two radii are called circular sectors (Fig. 1). A chord - a segment connecting two points on a circle - divides the circle into two segments, and the circle into two arcs (Fig. 2). A perpendicular drawn from the center to the chord divides it and the arcs subtended by it in half. The chord is longer, the closer it is located to the center; the longest chords - the chords passing through the center - are called diameters (of a circle, circle).

If a straight line is located at a distance $d,$ from the center of the circle, then for $d > R$ it does not intersect with the circle, for $d

Arcs of a circle, like angles, can be measured in degrees and fractions. A degree is taken to be $1/360$ of the entire circle. The central angle $AOB$ (Fig. 3) is measured in the same number of degrees as the arc $AB,$ on which it rests; the inscribed angle $ACB$ is measured by half the arc $AB.$ If the vertex $P$ of the angle $APB$ lies inside the circle, then this angle in degrees is equal to half the sum of the arcs $AB$ and $A′B′$ (Fig. 4, a ). The angle with the vertex $P$ outside the circle (Fig. 4, b), cutting out the arcs $AB$ and $A′B′,$ on the circle, is measured by the half-difference of the arcs $A′B′$ and $AB.$ Finally, the angle between the tangent and chord is equal to half the arc of a circle enclosed between them (Fig. 4, c).

A circle and a circle have an infinite number of axes of symmetry.

From the theorems on the measurement of angles and the similarity of triangles follow two theorems on proportional segments in a circle. The chord theorem says that if a point $M$ lies inside a circle, then the product of the lengths of the segments $AM⋅BM$ of chords passing through it is constant. In Fig. 5, and $AM⋅BM=A′M′⋅B′M.$ The secant and tangent theorem (meaning the lengths of segments - parts of these lines) states that if the point $M$ lies outside the circle, then the product of the secant $ MA$ to its outer part $MB$ is also unchanged and equal to the square of the tangent $MC$ (Fig. 5, b).

Even in ancient times, they tried to solve problems related to the circle - to measure the length of a circle or its arc, the area of a circle or sector, segment. The first of them has a purely “practical” solution: you can lay a thread along a circle, and then unroll it and attach it to a ruler, or mark a point on the circle and “roll” it along the ruler (you can, on the contrary, “roll” a circle with a ruler). One way or another, measurements showed that the ratio of the circumference $L$ to its diameter $d=2R$ is the same for all circles. This ratio is usually denoted by the Greek letter $π$ (“pi” is the initial letter of the Greek word perimetron, which means “circle”).

The solution was found as follows: if we consider regular $n$-gons $M_n,$ inscribed in a circle $K$, then as $n,$ tends to infinity, $M_n$ in the limit tends to $K.$ Therefore, it is natural to introduce the following, already strict definitions: the circumference $L$ is the limit of the sequence of perimeters $P_n$ of regular $n$-gons inscribed in the circle, and the area of the circle $S$ is the limit of the sequence $S_n$ of their areas. This approach is also accepted in modern mathematics, and in relation not only to the circle and circle, but also to other curved areas or areas limited by curvilinear contours: instead of regular polygons, sequences of broken lines with vertices on curves or contours of areas are considered, and the limit is taken when the length tends to the greatest links of the broken line to zero.

The length of a circular arc is determined in a similar way: the arc is divided into n equal parts, the division points are connected by a broken line, and the length of the arc $L$ is set equal to the limit of the perimeters $l_n$ of such broken lines as $n,$ tends to infinity. (Like the ancient Greeks, we do not clarify the concept of limit itself - it no longer refers to geometry and was quite strictly introduced only in the 19th century.)

From the very definition of the number π the formula for the circumference follows:

For the arc length, we can write a similar formula: since for two arcs $Γ$ and $Γ′$ with a common central angle, similarity considerations imply the proportion $l_n:l′_n=R:R′,$ and from this the proportion $l_n: R=l′_n:R′,$ after passing to the limit we obtain the independence (of the arc radius) of the relation $l/R=l′/R′=α.$ This relation is determined only by the central angle $AOB$ and is called the radian measure this angle and all arcs corresponding to it with center at $O.$ Thus, we obtain a formula for the length of the arc:

where $α$ is the radian measure of the arc.

The written formulas for $L$ and $l$ are just rewritten definitions or notations, but with their help we obtain formulas for the areas of a circle and a sector that are far from just notations:

$S=πR^2,$ $S=\frac(1)(2)αR^2.$

To derive the first formula, it is enough to go to the limit in the formula for the area of a regular n-gon inscribed in a circle:

$S_n=\frac(1)(2)P_nh_n.$

By definition, the left side tends to the area of the circle $S,$ and the right side tends to the number

$\frac(1)(2)LR=\frac(1)(2)⋅2πR⋅R =πR^2$

(the apothem $h_n,$ of course tends to $R$). The formula for the area of the sector $s$ is derived in exactly the same way:

$s=\lim S_n=\lim (\frac(1)(2)l_nh_n)=$ $\frac(1)(2)\lim l_n⋅\lim h_n=$ $\frac(1)(2)lR =$ $\frac(1)(2)αR^2$

($\lim $- read “limit”). Thus, the problem of determining the area of a segment with chord $AB,$ is solved, since it is represented as the difference or sum (Fig. 1, 2) of the areas of the corresponding sector and triangle $AOB.$