Arbitrary circle. What is a circle and a circle, what are their differences and examples of these figures from life
To in in general terms imagine what a circle is, look at a ring or a hoop. You can also take a round glass and a cup, put it upside down on a piece of paper and circle it with a pencil. With multiple magnification, the resulting line will become thick and not quite even, and its edges will be blurry. The circle as a geometric figure does not have such a characteristic as thickness.
Circle: definition and main means of description
A circle is a closed curve consisting of many points located in the same plane and equidistant from the center of the circle. In this case, the center is in the same plane. As a rule, it is denoted by the letter O.
The distance from any of the points of the circle to the center is called the radius and is denoted by the letter R.
If you connect any two points of the circle, then the resulting segment will be called a chord. The chord passing through the center of the circle is the diameter, denoted by the letter D. The diameter divides the circle into two equal arcs and is twice the length of the radius. Thus D = 2R, or R = D/2.
chord properties
- If a chord is drawn through any two points of the circle, and then a radius or diameter is drawn perpendicular to the latter, then this segment will split both the chord and the arc cut off by it into two equal parts. The converse is also true: if the radius (diameter) divides the chord in half, then it is perpendicular to it.
- If two parallel chords are drawn within the same circle, then the arcs cut off by them, as well as enclosed between them, will be equal.
- Draw two chords PR and QS intersecting within a circle at point T. The product of the segments of one chord will always be equal to the product of the segments of the other chord, that is, PT x TR = QT x TS.
Circumference: general concept and basic formulas
One of basic characteristics given geometric figure is the circumference. The formula is derived using values such as radius, diameter, and the constant "π", which reflects the constancy of the ratio of the circumference of a circle to its diameter.
Thus, L = πD, or L = 2πR, where L is the circumference, D is the diameter, R is the radius.
The formula for the circumference of a circle can be considered as the initial formula for finding the radius or diameter for a given circumference: D = L/π, R = L/2π.
What is a circle: basic postulates
- do not have common points;
- have one common point, while the line is called a tangent: if you draw a radius through the center and the point of contact, then it will be perpendicular to the tangent;
- have two common points, and the line is called a secant.
2. Through three arbitrary points lying in the same plane, at most one circle can be drawn.
3. Two circles can touch only at one point, which is located on the segment connecting the centers of these circles.
4. For any rotation about the center, the circle goes into itself.
5. What is a circle in terms of symmetry?
- the same curvature of the line at any of the points;
- relative to point O;
- mirror symmetry about the diameter.
6. If you construct two arbitrary inscribed angles based on the same circular arc, they will be equal. The angle based on an arc equal to half, that is, cut off by a chord-diameter, is always equal to 90 °.
7. If we compare closed curved lines of the same length, it turns out that the circle delimits the section of the plane of the largest area.
A circle inscribed in a triangle and circumscribed about it
The idea of \u200b\u200bwhat a circle is will be incomplete without a description of the features of this relationship with triangles.
- When constructing a circle inscribed in a triangle, its center will always coincide with the intersection point of the triangle.
- The center of a circle circumscribed about a triangle is located at the intersection of the median perpendiculars to each of the sides of the triangle.
- If you describe a circle around, then its center will be in the middle of the hypotenuse, that is, the latter will be the diameter.
- The centers of the inscribed and circumscribed circles will be at the same point if the base for construction is
Basic statements about the circle and quadrilaterals
- A circle can be circumscribed around a convex quadrilateral only if the sum of its opposite internal corners equals 180°.
- It is possible to construct a circle inscribed in a convex quadrilateral if the sum of the lengths of its opposite sides is the same.
- It is possible to describe a circle around a parallelogram if its angles are right.
- A circle can be inscribed in a parallelogram if all its sides are equal, that is, it is a rhombus.
- It is possible to construct a circle through the angles of a trapezoid only if it is isosceles. In this case, the center of the circumscribed circle will be located at the intersection of the quadrilateral and the median perpendicular drawn to the side.
School time for most adults is associated with a carefree childhood. Of course, many are reluctant to attend school, but only there they can get the basic knowledge that will later be useful to them in life. One such is the question of whether and circle. It is quite easy to confuse these concepts, because the words are of the same root. But the difference between them is not as big as it might seem to an inexperienced child. Children love this theme because of its simplicity.
What is a circle?
A circle is a closed line, each point of which is equidistant from the center. by the most a prime example circle is a hoop, which is a closed body. Actually, there is no need to talk too 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 decide to colorize the circle drawn above. To do this, you can choose any colors: blue, yellow or green - whichever is closer to your liking. And so you began to fill the void with something. After this was completed, we got a figure called a circle. In fact, a circle is a part of the surface outlined by a circle.
The circle has several important parameters, some of which are also characteristic of the circle. The first is the radius. It is the distance between the center point of the circle (well, or circle) and the circle itself, which creates the boundaries of the circle. Second important characteristic, which is repeatedly used in school problems, is the diameter (that is, the distance between opposite points of the circle).
And finally, the third characteristic inherent in the circle is the 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 set a clear center through which to 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 kind of circle. For example, if you look at the wheel of the car directly, then here is an example of a finished circle. Yes, it does not have to be filled in one 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 it, but it looks like a small circle in the sky.
Yes, the Sun itself is not a circle, it also has volume. But the sun itself, which we see above our head in the summer, is a typical circle. True, he still cannot calculate the area. After all, its comparison with a circle is given only for clarity, so that it is easier to understand what a circle and a circle are.
Differences between a circle and a circle
So what conclusion can we draw? What distinguishes a circle from 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 seem sometimes that there is no circumference in a circle, but it is not. 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.
Also, the circle can be something that makes the circle stand out from the background. For example, in the image above, the blue circle is on a white background. But that line, by which we understand that the figure begins here, is called in this case a circle. So a circle 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 remember pizza. When it is cut into equal pieces, they are all sectors of the circle, which is represented as such delicious dish. In this case, the sectors do not have to be equal at all. They may be different sizes. For example, if you cut off half of the pizza, then it will also be a sector of this circle.
The object displayed by this concept can only have a circle. can also be drawn, of course, but after that it will become a circle) has no area, so the sector cannot be selected.
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. The student needs to be prepared for the fact that the circle is a capricious figure. But, as they say, hard in learning - easy in battle. Yes, geometry is a complex science. But the successful development of it allows you to take a small step towards success. Because the efforts in training allow not only to replenish the luggage of one's own knowledge, but also to acquire the skills necessary in life. In fact, this is what the school is about. And the answer to the question of what a circle and a circle are is secondary, albeit important.
Demo material: compass, experiment material: items round shape and ropes (for each student) and rulers; circle model, colored crayons.
Target: Studying the concept of "circle" and its elements, establishing a connection between them; introduction of new terms; formation of the ability to conduct observations and draw conclusions using experimental data; education of cognitive interest in mathematics.
During the classes
I. Organizational moment
Greetings. Goal setting.
II. Verbal counting
III. new material
Among all kinds of flat figures, two main ones stand out: a triangle and a circle. These figures are known to you from early childhood. How to define a triangle? Through cuts! How do you define a circle? After all, this line bends at every point! The famous mathematician Grathendieck, recalling his school years, noticed that he became interested in mathematics after he learned the definition of a circle.
Draw a circle using a geometric tool - compass. Construction of a circle with a demonstration compass on the board:
- mark a point on the plane;
- we combine the leg of the compass with the tip with the marked point, and rotate the leg with the stylus around this point.
The result is a geometric figure - circle.
(Slide #1)
So what is a circle?
Definition. Circumference - is a closed curved line, all points of which are at an equal distance from a given point of the plane, called center circles.
(Slide #2)
Into how many parts does the plane divide the circle?
Point O- Centre circles.
OR- radius circle (this is a segment connecting the center of the circle with any point on it). in latin radius- wheel spoke.
AB- chord circle (this is a line segment that connects any two points on the circle).
DC- diameter circle (this is a chord passing through the center of the circle). Diameter - from the Greek "diameter".
DR– arc circle (this is the part of the circle bounded by two points).
How many radii and diameters can be drawn in a circle?
Part of the plane inside the circle and the circle itself form a circle.
Definition. A circle - is the part of the plane bounded by the circle. The distance from any point on the circle to the center of the circle does not exceed the distance from the center of the circle to any point on the circle.
What is the difference between a circle and a circle, and what do they have in common?
How are the lengths of the radius (r) and diameter (d) of one circle related?
d=2*r (d is the length of the diameter; r- radius length)
How are the lengths of the diameter and any chord related?
Diameter is the largest of the chords of a circle!
The circle is an amazingly harmonious figure, the ancient Greeks considered it the most perfect, since the circle is the only curve that can “slide by itself”, revolving around the center. The basic property of a circle answers the questions why compasses are used to draw it and why wheels are made round, and not square or triangular. By the way, about the wheel. This is one of the greatest inventions of mankind. It turns out that thinking about the wheel was not as easy as it might seem. After all, even the Aztecs who lived in Mexico did not know the wheel until almost the 16th century.
The circle can be drawn on checkered paper without a compass, that is, by hand. True, the circle turns out to be a certain size. (The teacher shows on the checkered board)
The rule for drawing such a circle is written as 3-1, 1-1, 1-3.
Freehand draw a quarter of such a circle.
How many squares is the radius of this circle? They say that the great German artist Albrecht Dürer could draw a circle so accurately with one movement of his hand (without rules) that a subsequent check with a compass (the center was indicated by the artist) did not show any deviations.
Laboratory work
You already know how to measure the length of a segment, find the perimeters of polygons (triangle, square, rectangle). But how to measure the circumference of a circle, if the circle itself is a curved line, and the unit of length is a segment?
There are several ways to measure the circumference of a circle.
Circle trace (one turn) on a straight line.
The teacher draws a straight line on the blackboard, marks a point on it and on the border of the circle model. Aligns them, and then smoothly rolls the circle in a straight line until the marked point BUT on a circle will not be on a straight line at a point IN. Section AB then it will be equal to the circumference.
Leonardo da Vinci: "The movement of wagons has always shown us how to straighten the circumference of a circle."
Assignment to students:
a) draw a circle by circling the bottom of a round object;
b) wrap the bottom of the object with a thread (once) so that the end of the thread coincides with the beginning at the same point on the circle;
c) straighten this thread to a segment and measure its length using a ruler, this will be the circumference.
The teacher is interested in the measurement results of several students.
However, these methods of directly measuring the circumference are not very convenient and give roughly approximate results. Therefore, already from ancient times, they began to look for more advanced ways to measure the circumference of a circle. In the process of measurements, it was noticed that there is a certain relationship between the circumference of a circle and the length of its diameter.
d) Measure the diameter of the bottom of the object (the largest of the chords of the circle);
e) find the ratio С:d (up to tenths).
Ask a few students for the results of the calculations.
Many scientists - mathematicians tried to prove that this ratio is a constant number, independent of the size of the circle. For the first time this was done by the ancient Greek mathematician Archimedes. He found a fairly accurate value for this ratio.
This relationship began to be denoted by the Greek letter (read “pi”) - the first letter Greek word"periphery" - a circle.
C is the circumference;
d is the length of the diameter.
Historical information about the number π:
Archimedes, who lived in Syracuse (Sicily) from 287 to 212 BC, found the meaning without measurements, just by reasoning
In fact, the number π cannot be expressed by any exact fraction. The 16th-century mathematician Ludolph had the patience to calculate it with 35 decimal places and bequeathed to carve this value of π on his grave monument. In 1946 - 1947. two scientists independently calculated 808 decimal places for pi. Now more than a billion digits of the number π have been found on computers.
The approximate value of π with an accuracy of five decimal places can be remembered using the following line (according to the number of letters in a word):
π ≈ 3.14159 – “I know this and remember it perfectly”.
Introduction to the formula for the circumference of a circle
Knowing that C:d \u003d π, what will be the length of the circle C?
(Slide #3) C = πd C = 2πr
How did the second formula come about?
Reads: circumference is equal to the product of the number π by its diameter (or twice the product of the number π by its radius).
Area of a circle is equal to the product of the number π and the square of the radius.
S= πr2
IV. Problem solving
№1. Find the length of a circle whose radius is 24 cm. Round the number π to hundredths.
Solution:π ≈ 3.14.
If r = 24 cm, then C = 2 π r ≈ 2 3.14 24 = 150.72(cm).
Answer: circumference 150.72 cm.
No. 2 (oral): How to find the length of an arc equal to a semicircle?
A task: If you wrap Earth along the equator with a wire and then add 1 meter to its length, can a mouse slip between the wire and the ground?
Solution: C \u003d 2 πR, C + 1 \u003d 2 π (R + x) 
Not only a mouse, but also a large cat will slip into such a gap. And it would seem, what does 1 m mean compared to 40 million meters of the earth's equator?
V. Conclusion
- What are the main points to pay attention to when constructing a circle?
- What parts of the lesson were the most interesting for you?
- What new did you learn in this lesson?
Picture crossword solution(Slide #3)
It is accompanied by a repetition of the definitions of a circle, chord, arc, radius, diameter, formulas for the circumference. And as a result - the keyword: "CIRCLE" (horizontally).
Lesson summary: grading, comments on performance homework.Homework: p. 24, No. 853, 854. Conduct an experiment to find the number π 2 more times.
AND a circle - geometric figures, interconnected. there is a boundary polyline (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 chosen, taken as the center of the circle, and a closed line is drawn using a compass.
If the 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 large letter "er" ( r or R). There are as many radii in a circle as there are points in the circumference.
A line segment connecting two points of a circle and passing through its center is called a diameter. Diameter consists of two radii lying on the same straight line. The diameter is indicated by the Latin small or large letter "de" ( d or D).
Rule. Diameter circle is equal to two of its radii.
d = 2r
D=2R
The circumference 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 of a circle is greater than its diameter. The number ¶ has an infinite number of decimal places. For calculations it is accepted ¶ = 3.14.
The circumference of a circle is denoted by the Latin capital letter "ce" ( C). The circumference of a circle is proportional to its diameter. Formulas for calculating the circumference of a circle by its radius and diameter:
C = ¶d
C = 2r
- Examples
- Given: d = 100 cm.
- Circumference: C=3.14*100cm=314cm
- Given: d = 25 mm.
- Circumference: C=2*3.14*25=157mm
The secant of the circle and the arc of the circle
Any secant (straight line) intersects the 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 large and small 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 the secant passes through the center of the 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 more - 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 that lies inside a circle.
The center, radius, diameter of a circle are at the same time the center, radius and diameter of the corresponding circle.
Since a circle is part of a plane, one of its parameters is the area.
Rule. Area of a circle ( S) is equal to the product of the square of the radius ( r2) to the number ¶.
- Examples
- Given: r = 100 cm
- Area of a circle:
- S \u003d 3.14 * 100 cm * 100 cm \u003d 31,400 cm 2 ≈ 3m 2
- Given: d = 50 mm
- Area of a circle:
- S \u003d ¼ * 3.14 * 50 mm * 50 mm \u003d 1 963 mm 2 ≈ 20 cm 2
If two radii are drawn in a circle different points circle, then two parts of the circle are formed, which are called sectors. If a chord is drawn in a circle, then the part of the plane between the arc and the chord is called circle segment.