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. A prerequisite for a circle is that 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, the boundary of which consists of a large number of 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 given 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 is easy to verify if you draw a radius through this point (Fig. 85).
A point taken outside the circle is located from its center at a distance greater than the radius. 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 straight 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.
Demo material: compass, material for experiment: round objects and ropes (for each student) and rulers; circle model, colored crayons.
Target: Studying the concept of “circle” and its elements, establishing connections between them; introduction of new terms; developing the ability to make observations and draw conclusions using experimental data; nurturing cognitive interest in mathematics.
During the classes
I. Organizational moment
Greetings. Setting a goal.
II. Verbal counting
III. New material
Among all kinds of flat figures, two main ones stand out: the triangle and the circle. These figures have been known to you since early childhood. How to define a triangle? Through segments! How can we determine what a circle is? After all, this line bends at every point! The famous mathematician Grathendieck, recalling his school years, noted that he became interested in mathematics after learning the definition of a circle.
Let's draw a circle using a geometric device - compass. Constructing a circle with a demonstration compass on the board:
- mark a point on the plane;
- We align 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 No. 1)
So what is a circle?
Definition. Circumference - is a closed curved line, all points of which are at equal distances from a given point on the plane, called center circles.
(Slide No. 2)
How many parts does a plane divide a circle into?
Point O- center 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 segment connecting any two points on a circle).
DC – diameter circle (this is a chord passing through the center of the circle). Diameter comes from the Greek “diameter”.
DR– arc circle (this is a part of a circle bounded by two points).
How many radii and diameters can be drawn in a circle?
The part of the plane inside the circle and the circle itself form a circle.
Definition. Circle - This is the part of the plane bounded by a 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.
How do a circle and a circle differ from each other, and what do they have in common?
How are the lengths of the radius (r) and diameter (d) of one circle related to each other?
d = 2 * r (d– diameter length; r – radius length)
How are the lengths of a 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 on its own”, rotating around the center. The main 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 coming up with 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. (Teacher shows on the checkered board)
The rule for depicting such a circle is written as 3-1, 1-1, 1-3.
Draw a quarter of such a circle by hand.
How many cells 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). How to measure the length of a circle if the circle itself is a curved line, and the unit of measurement of length is a segment?
There are several ways to measure circumference.
The trace from the circle (one revolution) on a straight line.
The teacher draws a straight line on the board, marks a point on it and on the boundary of the circle model. Combines them, and then smoothly rolls the circle in a straight line until the marked point A on a circle will not be on a straight line at a point IN. Line segment AB will then be equal to the circumference.
Leonardo da Vinci: "The movement of carts 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 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 inconvenient and give rough results. Therefore, since ancient times, they began to look for more advanced ways to measure circumference. During the measurement process, we noticed that there is a certain relationship between the length 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 C:d (accurate to tenths).
Ask several students for the results of calculations.
Many scientists and mathematicians tried to prove that this ratio is a constant number, independent of the size of the circle. The ancient Greek mathematician Archimedes was the first to do this. He found a fairly accurate meaning for this ratio.
This relationship began to be denoted by a Greek letter (read “pi”) - the first letter of the Greek word “periphery” is a circle.
C – circumference;
d – diameter length.
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 as an exact fraction. The 16th century mathematician Ludolph had the patience to calculate it with 35 decimal places and bequeathed this value of π to be carved on his grave monument. In 1946 – 1947 two scientists independently calculated the 808 decimal places of pi. Now more than a billion digits of the number π have been found on computers.
The approximate value of π, accurate to five decimal places, can be remembered using the following line (based on the number of letters in the word):
π ≈ 3.14159 – “I know and remember this perfectly.”
Introduction to the Circumference Formula
Knowing that C:d = π, what will be the length of circle C?
(Slide No. 3) C = πd C = 2πr
How did the second formula come about?
Reads: circumference is equal to the product of the number π and its diameter (or twice the product of the number π and its radius).
Area of a circle is equal to the product of the number π and the square of the radius.
S= πr 2
IV. Problem solving
№1. Find the length of a circle whose radius is 24 cm. Round the number π to the nearest hundredth.
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 (orally): How to find the length of an arc equal to a semicircle?
Task: If you wrap a wire around the globe along the equator and then add 1 meter to its length, will a mouse be able to slip between the wire and the ground?
Solution: C = 2 πR, C+1 = 2π(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 main points should you pay attention to when constructing a circle?
- What parts of the lesson were most interesting to you?
- What new did you learn in this lesson?
Solution to crossword puzzle with pictures(Slide No. 3)
It is accompanied by a repetition of the definitions of circle, chord, arc, radius, diameter, formulas for circumference. And as a result - the keyword: “CIRCLE” (horizontally).
Lesson summary: grading, comments on homework. Homework: p. 24, No. 853, 854. Conduct an experiment to find the number π 2 more times.
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 striking example of a 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.
A 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. The second important characteristic, which is repeatedly used in school problems, is 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 difficult to look at it, 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 image above, the blue circle is on a white background. But the line by which we understand that the figure begins here is called in this case a 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 presented in the form of such a 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.