Circle. Shape and position. What is a circle and a circle, what are their differences and examples of these figures from life

Let's understand what a circle and a circle are. Formula for area of ​​a circle and circumference.

Every day we come across many objects that are shaped like a circle or, on the contrary, a circle. Sometimes the question arises what a circle is and how it differs from a circle. Of course, we've all taken geometry lessons, but sometimes it doesn't hurt to brush up on your knowledge with some very simple explanations.

What is the circumference and area of ​​a circle: definition

So, a circle is a closed curved line that limits or, on the contrary, forms a circle. Required condition circle - it has a center and all points are equidistant from it. Simply put, a circle is a gymnastics hoop (or as it is often called a hula hoop) on a flat surface.

The circumference of a circle is the total length of the very curve that forms the circle. As is known, regardless of the size of the circle, the ratio of its diameter and length is equal to the number π = 3.141592653589793238462643.

It follows from this that π=L/D, where L is the circumference and D is the diameter of the circle.

If you know the diameter, then the length can be found using a simple formula: L= π* D

If the radius is known: L=2 πR

We have figured out what a circle is and can move on to the definition of a circle.

A circle is a geometric figure that is surrounded by a circle. Or, a circle is a figure whose boundary consists of large quantity points equidistant from the center of the figure. The entire area that is inside a circle, including its center, is called a circle.

It is worth noting that the circle and the circle that is located in it have the same radius and diameter. And the diameter, in turn, is twice as large as the radius.

A circle has an area on a plane, which can be found using a simple formula:

Where S is the area of ​​the circle, and R is the radius of the circle.

How does a circle differ from a circle: explanation

The main difference between a circle and a circle is that a circle is a geometric figure, while a circle is a closed curve. Also note the differences between a circle and a circle:

• A circle is a closed line, and a circle is the area within that circle;
• A circle is a curved line on a plane, and a circle is a space closed into a ring by a circle;
• Similarities between circle and circle: radius and diameter;
• The circle and the circumference have a single center;
• If the space inside the circle is shaded, it turns into a circle;
• A circle has a length, but a circle does not, and vice versa, a circle has an area, which a circle does not.

Circle and circumference: examples, photos

For clarity, we suggest looking at a photo that shows a circle on the left and a circle on the right.

Formula for circumference and area of ​​a circle: comparison

Formula for circumference L=2 πR

Formula for the area of ​​a circle S= πR²

Please note that both formulas contain the radius and the number π. It is recommended to memorize these formulas, as they are the simplest and will definitely come in handy in Everyday life and at work.

Area of ​​a circle by circumference: formula

S=π(L/2π)=L²/4π, where S is the area of ​​the circle, L is the circumference.

Video: What is a circle, circumference and radius

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

The part of the plane bounded by a circle is called a circle.

A straight line segment connecting a point on a circle with its center is called a radius(Fig. 84).

Since all points of the circle are at the same distance from the center, then all radii of the same circle are equal to each other. The radius is usually denoted by the letter R or r.

A point taken inside a circle is located from its center at a distance less than the radius. This can be easily verified if this point draw the radius (Fig. 85).

A point taken outside the circle is located at a distance greater than the radius from its center. This can be easily verified by connecting this point to the center of the circle (Fig. 85).

A straight line segment connecting two points on a circle is called a chord.

The chord passing through the center is called the diameter(Fig. 84). The diameter is usually denoted by the letter D. The diameter is equal to two radii:

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

Theorem. A chord that does not pass through the center of a circle is smaller than the diameter drawn in the same circle.

In fact, if we draw some chord, for example AB, and connect its ends with the center O (Fig. 86), we will see that the chord AB is smaller than the broken line AO ​​+ OB, i.e. AB r, and since 2 r= D, then AB

If the circle is bent along the diameter (Fig. 87), then both parts of the circle and the circle will align. The diameter divides the circle and circumference into two equal parts.

Two circles (two circles) are called equal if they can be superimposed on each other so that they coincide.

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

2. Arc of a circle.

Part of a circle is called an arc.

The word "arc" is sometimes replaced by the sign \(\breve( )\). An arc is designated by two or three letters, two of which are placed at the ends of the arc, and the third at some point on the arc. In drawing 88, two arcs are indicated: \(\breve(ACB)\) and \(\breve(ADB)\).

When an arc is smaller than a semicircle, it is usually denoted by two letters. Thus, arc ADB can be designated \(\breve(AB)\) (Fig. 88). A chord that connects the ends of an arc is said to subtend the arc.

If we move the arc AC (Fig. 89, a) so that it slides along the given circle, and if at the same time it coincides with the arc MN, then \(\breve(AC)\) = \(\breve(NM)\).

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

Therefore \(\breve(AC)\) > \(\breve(AB)\); \(\breve(AB)\)

Constructing a circle using three points

Task. Draw a circle through three points that do not lie on the same line.

Let us be given three points A, B and C that do not lie on the same straight line (Fig. 311).

Let's connect these points with segments AB and BC. To find points equidistant from points A and B, divide the segment AB in half and draw a line perpendicular to AB through the middle (point M). Each point of this perpendicular is equally distant from points A and B.

To find points equidistant from points B and C, we divide the segment BC in half and draw a line perpendicular to BC through its middle (point N). Each point of this perpendicular is equally distant from points B and C.

Point O of the intersection of these perpendiculars will be at the same distance from these points A, B and C (AO = BO = CO). If we, taking point O as the center of a circle, with a radius equal to AO, draw a circle, then it will pass through all given points A, B and C.

Point O is the only point that can serve as the center of a circle passing through three points A, B and C that do not lie on the same line, since two perpendiculars to segments AB and BC can intersect only at one point. This means that the problem has a unique solution.

Note. If three points A, B and C lie on the same line, then the problem will not have a solution, since the perpendiculars to the segments AB and BC will be parallel and there will be no point equally distant from points A, B, C, i.e. . a point that could serve as the center of the desired circle.

If we connect points A and C with a segment and connect the middle of this segment (point K) with the center of the circle O, then OK will be perpendicular to AC (Fig. 311), since in the isosceles triangle AOC OK is the median, therefore OK⊥AC.

Consequence. Three perpendiculars to the sides of a triangle drawn through their midpoints intersect at one point.

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:

1. Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
2. 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 find that the square of the length of the tangent segment will be equal to the product the entire segment secant to 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 the circle and located between two chords is identical to half the sum of the angular values ​​of the arcs of the circle that are contained within 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 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 intersect internal corners figure, 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.

There is the following 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.

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

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

R = \frac(abc)(4 S)

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 a cyclic quadrilateral.

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

For most adults, school time is associated with a carefree childhood. Of course, many are reluctant to attend school, but only there can they gain basic knowledge that will later be useful to them in life. One of these is the question of whether and the circle. It is quite easy to confuse these concepts, because the words have the same root. But the difference between them is not as big as it might seem to an inexperienced child. Children love this topic because of its simplicity.

What is a circle?

A circle is a closed line, each point of which is equally distant from the central one. The most a shining example The circle is a hoop, which is a closed body. Actually, there is no need to talk much about the circle. In the question of what a circle and a circle are, its second part is much more interesting.

What is a circle?

Imagine that you decided to color the circle drawn above. To do this, you can choose any colors: blue, yellow or green - whatever suits your taste. And so you began to fill the void with something. Once this was completed, we ended up with a shape called a circle. Essentially, a circle is a part of a surface outlined by a circle.

The circle has several important parameters, some of which are also characteristic of a circle. The first is the radius. It is the distance between the central point of a circle (or circle) and the circle itself, which creates the boundaries of the circle. Second important characteristic, which is used repeatedly in school tasks- this is the diameter (that is, the distance between opposite points of the circle).

And finally, the third characteristic inherent in a circle is area. This property is specific only to it, the circle has no area due to the fact that it has nothing inside, and the center, unlike the circle, is more imaginary than real. In the circle itself, you can establish a clear center through which you can draw a series of lines that divide it into sectors.

Examples of a circle in real life

In fact, there are enough possible objects that can be called a type of circle. For example, if you look directly at a car wheel, then here is an example of a finished circle. Yes, it does not have to be filled in a single color; various patterns inside it are quite possible. The second example of a circle is the sun. Of course, it will be hard to look at, but it looks like a small circle in the sky.

Yes, the Sun star itself is not a circle, it also has volume. But the sun itself, which we see above our heads in the summer, is a typical circle. True, he still won’t be able to calculate the area. After all, its comparison with a circle is given only for clarity, to make it easier to understand what a circle and a circle are.

Differences between a circle and a circle

So what conclusion can we draw? The difference between a circle and a circle is that the latter has an area, and in most cases the circle is the boundary of the circle. Although there are exceptions at first glance. It may sometimes seem that there is no circle in a circle, but this is not so. In any case, there is something. It’s just that the circle can be very small, and then it is not visible to the naked eye.

The circle can also be what makes the circle stand out from the background. For example in the above image blue circle is on a white background. But the line by which we understand that the figure begins here is called in this case circle. Thus, the circumference is a circle. This is the difference between a circle and a circle.

What is a sector?

A sector is a section of a circle that is formed by two radii drawn along it. To understand this definition, you just need to think about pizza. When it is cut into equal pieces, all of them are sectors of the circle, which is represented in the form of this delicious dish. In this case, the sectors do not necessarily have to be equal. They can be of different sizes. For example, if you cut half of a pizza, it will also be a sector of this circle.

The object represented by this concept can only have a circle. This can also be done, of course, but after that it will become a circle) has no area, so it will not be possible to select a sector.

conclusions

Yes, the topic of circle and circumference (what is it) is very easy to understand. But in general, everything related to these is the most difficult to study. A student needs to be prepared for the fact that a circle is a capricious figure. But, as they say, it’s hard to learn, but it’s easy to fight. Yes, geometry is a complex science. But its successful mastery allows you to take a small step towards success. Because efforts in learning allow you not only to replenish your own knowledge, but also to acquire the skills necessary in life. Actually, this is what the school is aimed at. And the answer to the question of what a circle and a circle are is secondary, although important.

AND circle - geometric figures, 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 as 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 farther 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.