A geometric figure is called flat if all the fine parts of the figure belong to the same plane.
Examples of flat geometric shapes are: straight line, segment, circle, various polygons, etc. Shapes such as a ball, cube, cylinder, pyramid, etc. are not flat.
On a plane, convex and non-convex figures are distinguished.
A geometric figure is called convex if it entirely contains a segment whose ends are any two points belonging to the figure (Fig. 54).
Examples of convex shapes are: circle, various triangles, square. A point, straight line, ray, segment, plane are also considered convex figures.
The main geometric figures on a plane are a point and a straight line. These terms are often used even when working with preschoolers. It is necessary to promptly teach children to recognize these figures, depict them, understand and correctly complete tasks.
The basic properties of points and lines are revealed in the axioms:
1. There are points that belong and do not belong to a line.
2. A single straight line can be drawn through two different points.
3. Two different lines either do not intersect or intersect at one point.
Children, for example, in the process of playing or drawing, become familiar with a point, a segment, various lines, distinguishing from them a straight line, a curve, a broken line, and learn to recognize some of their properties.
1. “Which road from the forest to the house is shorter?” (Fig. 55).
2. “The piglets live in houses located on the banks of the river. They don't know how to swim. Which of the piglets can go visit each other?” (Fig. 56).
A closed line divides the plane into outer and inner regions. Children learn early what “in” and “out” mean. For example, this happens when performing the task of painting a figure, that is, its internal area.
The geometric shapes that children become familiar with early (circle, square, triangle, etc.) are closed lines (the boundaries of the shapes) with their inner region. Circle border
is a circle. The border of polygons is a broken line, which consists of segments. In geometry, all these concepts have definitions.
A segment is a part of a line that consists of all points of this line lying between two given points, called the ends of the segment.
A ray (half-line) is a part of a line, consisting of all its points lying on one side of a given point on it (the beginning of the ray).
An angle is a smaller part of a plane bounded by two rays emanating from one point. These rays are called the sides of the angle, and their common point is the vertex of the angle (Fig. 59).
A circle can be defined as a figure consisting of a circle and its inner region.
Circle is a set of points on a plane equidistant from a given point. This point O is called the center of the circle, and the given distance R is its radius (Fig. 64).
IN kindergarten children are also introduced to the oval (“a figure similar to a circle in that it has no corners or sides, but differs from a circle in its elongation”). In geometry, such a term is not considered, but the ellipse is studied. It is not advisable to offer it to children due to the complexity of its construction. Since the words “oval”, “object of an oval shape” are often used in everyday life, knowledge about the oval is necessary for children as an element of sensory education and speech development.
Polygons
Polygon- part of the plane bounded by a simple closed broken line. The links of the polygon are called the sides of the polygon, and the vertices are called vertices of the polygon. The boundary of a polygon (a simple closed polyline) is also called a polygon.
When working with preschoolers, models of figures made of cardboard, plastic or wood are usually considered, and tasks are offered on drawing polygons using stencils and outlines, and painting figures. In the process of this activity, children become familiar with the names of figures, their structure and some properties, use terms such as: the border of a figure, the internal region of a figure, etc.
A convex polygon lies in one half-plane relative to any straight line containing its side (Fig. 65).
The shape of a circle is interesting from the point of view of occultism, magic and the ancient meanings attached to it by people. All the smallest components around us - atoms and molecules - have a round shape. The sun is round, the moon is round, our planet is also round. Water molecules - the basis of all living things - also have a round shape. Even nature creates its life in circles. For example, you can remember about a bird's nest - birds also make it in this form.
This figure in the ancient thoughts of cultures
The circle is a symbol of unity. It is present in different cultures in many the smallest details. We don't even attach as much importance to this form as our ancestors did.
Since ancient times, a circle has been a sign of an endless line, which symbolizes time and eternity. In pre-Christian times it was the ancient sign of the wheel of the sun. All points in are equivalent, the line of a circle has neither beginning nor end.
And the center of the circle was the source of endless rotation of space and time for the Masons. The circle is the end of all figures; it is not for nothing that the secret of creation was contained in it, according to the Freemasons. The shape of the watch dial, which also has this shape, denotes an indispensable return to the point of departure.
This figure has a deep magical and mystical composition, which has been endowed by many generations of people from different cultures. But what is a circle as a figure in geometry?
What is a circle
The concept of a circle is often confused with the concept of a circle. This is no wonder, because they are very closely interconnected. Even their names are similar, which causes a lot of confusion in the immature minds of schoolchildren. To figure out “who is who,” let’s look at these questions in more detail.
By definition, a circle is a curve that is closed, and each point of which is equidistant from a point called the center of the circle.
What you need to know and what you can use to build a circle
To construct a circle, it is enough to select an arbitrary point, which can be designated as O (this is how the center of the circle is called in most sources, we will not deviate from traditional notations). The next step is to use a compass - a drawing tool, which consists of two parts with either a needle or a writing element attached to each of them.
These two parts are connected to each other by a hinge, which allows you to choose an arbitrary radius within certain limits related to the length of these same parts. With the help of this device, the tip of a compass is installed at an arbitrary point O, and a curve is already outlined with a pencil, which ultimately turns out to be a circle.
What are the dimensions of a circle?
If we connect the center of the circle and any arbitrary point on the curve obtained as a result of working with a compass using a ruler, we get All such segments, called radii, will be equal. If we connect two points on the circle and the center with a straight line using a ruler, we get its diameter.
A circle is also characterized by the calculation of its length. To find it, you need to know either the diameter or the radius of the circle and use the formula presented in the figure below.
In this formula, C is the circumference, r is the radius of the circle, d is the diameter, and Pi is a constant with a value of 3.14.
By the way, the constant Pi was calculated just from the circle.
It turned out that no matter what the diameter of the circle, the ratio of the circumference to the diameter is the same, equal to approximately 3.14.
What is the main difference between a circle and a circle?
Essentially, a circle is a line. It is not a figure, it is a curved closed line that has neither end nor beginning. And the space that is located inside it is emptiness. The simplest example of a circle is a hoop or, in other words, a hula hoop, which children use in class. physical culture or adults, in order to create a slender waist.
Now we come to the concept of what a circle is. This is first of all a figure, that is, a certain set of points, limited by line. In the case of a circle, this line is the circle discussed above. It turns out that a circle is a circle in the middle of which there is not emptiness, but many points in space. If we stretch fabric over a hula hoop, we will no longer be able to spin it, because it will no longer be a circle - its emptiness is replaced by fabric, a piece of space.
Let's move directly to the concept of a circle
Circle - geometric figure, which is part of a plane bounded by a circle. It is also characterized by such concepts as radius and diameter, discussed above when defining a circle. And they are calculated in exactly the same way. The radius of a circle and the radius of a circle are identical in size. Accordingly, the length of the diameter is also similar in both cases.
Since a circle is part of a plane, it is characterized by the presence of area. You can calculate it again using the radius and Pi. The formula looks like this (see picture below).
In this formula, S is the area, r is the radius of the circle. Pi is the same constant again, equal to 3.14.
The circle formula, which can also be calculated using the diameter, changes and takes the form shown in the following figure.
One-fourth comes from the fact that the radius is 1/2 the diameter. If the radius is squared, it turns out that the relationship is transformed to the form:
r*r = 1/2*d*1/2*d;
A circle is a figure in which individual parts, for example a sector, can be distinguished. It looks like part of a circle, which is limited by an arc segment and its two radii drawn from the center.
The formula that allows you to calculate the area of a given sector is presented in the figure below.
Using shapes in polygon problems
Also, a circle is a geometric figure that is often used in conjunction with other figures. For example, such as a triangle, trapezoid, square or rhombus. There are often problems where you need to find the area of an inscribed circle or, conversely, one circumscribed around a certain figure.
An inscribed circle is one that touches all sides of the polygon. The circle must have a point of contact with each side of any polygon.
For a certain type of polygon, the determination of the radius of the inscribed circle is calculated by separate rules, which are clearly explained in the geometry course.
We can cite a few of them as examples. The formula for a circle inscribed in polygons can be calculated as follows (several examples are shown in the photo below).
A few simple real-life examples to reinforce your understanding of the difference between a circle and a circle
Before us If it is open, then the iron edge of the hatch is a circle. If it is closed, then the lid acts as a circle.
A circle can also be called any ring - gold, silver or jewelry. The ring that holds a bunch of keys is also a circle.
But a round magnet on the refrigerator, a plate or pancakes baked by grandma are a circle.
The neck of a bottle or jar when viewed from above is a circle, but the lid that closes this neck is a circle when viewed from above.
There are many such examples that can be given, and in order to assimilate such material, they need to be given so that children better grasp the connection between theory and practice.
The circle, its parts, their sizes and relationships are things that a jeweler constantly encounters. Rings, bracelets, castes, tubes, balls, spirals - a lot of round things have to be made. How can you calculate all this, especially if you were lucky enough to skip geometry classes at school?..
Let's first look at what parts a circle has and what they are called.
- A circle is a line that encloses a circle.
- An arc is a part of a circle.
- Radius is a segment connecting the center of a circle with any point on the circle.
- A chord is a segment connecting two points on a circle.
- A segment is a part of a circle bounded by a chord and an arc.
- A sector is a part of a circle bounded by two radii and an arc.
The quantities we are interested in and their designations:
Now let's see what problems related to parts of a circle have to be solved.
- Find the length of the development of any part of the ring (bracelet). Given the diameter and chord (option: diameter and central angle), find the length of the arc.
- There is a drawing on a plane, you need to find out its size in projection after bending it into an arc. Given the arc length and diameter, find the chord length.
- Find out the height of the part obtained by bending a flat workpiece into an arc. Source data options: arc length and diameter, arc length and chord; find the height of the segment.
Life will give you other examples, but I gave these only to show the need to set some two parameters to find all the others. This is what we will do. Namely, we will take five parameters of the segment: D, L, X, φ and H. Then, choosing all possible pairs from them, we will consider them as initial data and find all the rest by brainstorming.
In order not to unnecessarily burden the reader, I will not give detailed solutions, but will present only the results in the form of formulas (those cases where there is no formal solution, I will discuss along the way).
And one more note: about units of measurement. All quantities, except the central angle, are measured in the same abstract units. This means that if, for example, you specify one value in millimeters, then the other does not need to be specified in centimeters, and the resulting values will be measured in the same millimeters (and areas in square millimeters). The same can be said for inches, feet and nautical miles.
And only the central angle in all cases is measured in degrees and nothing else. Because, as a rule of thumb, people who design something round don't tend to measure angles in radians. The phrase “angle pi by four” confuses many, while “angle forty-five degrees” is understandable to everyone, since it is only five degrees higher than normal. However, in all formulas there will be one more angle - α - present as an intermediate value. In meaning, this is half the central angle, measured in radians, but you can safely not delve into this meaning.
1. Given the diameter D and arc length L
; chord length 
;
segment height 
; central angle 
.
2. Given diameter D and chord length X
; arc length ;
segment height ; central angle 
.
Since the chord divides the circle into two segments, this problem has not one, but two solutions. To get the second, you need to replace the angle α in the above formulas with the angle .
3. Given the diameter D and central angle φ
; arc length ;
chord length ; segment height .
4. Given the diameter D and height of the segment H
; arc length ;
chord length ; central angle .
6. Given arc length L and central angle φ
; diameter ;
chord length ; segment height .
8. Given the chord length X and the central angle φ
; arc length 
;
diameter ; segment height .
9. Given the length of the chord X and the height of the segment H
; arc length ;
diameter ; central angle .
10. Given the central angle φ and the height of the segment H
; diameter 
;
arc length ; chord length .
The attentive reader could not help but notice that I missed two options:
5. Given arc length L and chord length X
7. Given the length of the arc L and the height of the segment H
These are just those two unpleasant cases when the problem does not have a solution that could be written in the form of a formula. And the task is not so rare. For example, you have a flat piece of length L, and you want to bend it so that its length becomes X (or its height becomes H). What diameter should I take the mandrel (crossbar)?
This problem comes down to solving the equations: 
; - in option 5
; - in option 7
and although they cannot be solved analytically, they can be easily solved programmatically. And I even know where to get such a program: on this very site, under the name . Everything that I am telling you here at length, she does in microseconds.
To complete the picture, let’s add to the results of our calculations the circumference and three area values - circle, sector and segment. (Areas will help us a lot when calculating the mass of all round and semicircular parts, but more on this in a separate article.) All these quantities are calculated using the same formulas:
circumference ;
area of a circle 
;
sector area 
;
segment area 
;
And in conclusion, let me remind you once again about the existence of absolutely free program, which performs all of the above calculations, freeing you from having to remember what an arctangent is and where to look for it.
Today we will make chicken. What color is the chicken? That's right, yellow. From all the circles, select only the yellow circles. Then set aside the blue circles and the green ones separately.
First, we simply lay out the chicken on paper without glue so that the baby has an understanding of what we are doing, this will also help to avoid mistakes when working with glue.
The large yellow circle will be the body of the chicken. Where do we put it? (we invite the child to choose a place on a piece of paper himself).
The smaller circle will be the head. Where will our chicken's head be? (let the child again choose the place in which direction the chicken will look: up at the sky and the sun or down at the grass, maybe he will peck the grains. Help the child to fantasize, offer options. You can give little ones a hint, advise, but don’t insist, let him will make his own choice)
Where is the little black circle? This will be the eye. A small triangle is the beak, two identical triangles are the paws. Place the figures in their places.
What is our chicken missing? That's right, wings! We have 2 more yellow circles, we will put one aside - this will be the sun, and from the second we will make wings. How do you think about making two wings from one circle? (children from three years old can handle this. Let the child hold the circle in his hands, turn it, apply it to the paper, perhaps he will come up with an answer).
We'll cut the circle in half. To do this, let's find the center of the circle. Where is the center (middle) of the circle? (you can give the child a pencil and offer to find and mark the center on the back (not colored!) side of the sheet. Even if the point is not in the center, but somewhere nearby, it’s okay, praise the baby! If the child is small, do everything yourself, explaining every action).
Now we will draw a straight line through the center, which will divide the circle in half. Along this line we will cut our circle into two parts. You get two wings (be sure to cut through the point (center) indicated by the child, firstly, the child will feel that his opinion is important to you and you listen to him, and secondly, the applique will be more artistic)
During a lesson for older children, you can explain what a semicircle is (or remember this figure)
Look at the shapes we got. This figure is called a semicircle. Half a circle - semicircle (repeat several times and suggest repeating the name)
Where will our chicken's wings be?
The chicken was laid out on paper, now you can glue it.
The chicken is ready.
Let's take large green circles (or 1 circle) - this will be our grass. How do you think about making grass from a circle? That's right, cut in half again (we repeat the steps as with the wings: let the child mark the center, cut and glue at the bottom). To make the grass more natural, you can make small cuts along the rounded side.
Glue the sun to the sky.
Clouds can be made different ways:
1. Glue the circles overlapping, forming a cloud. Different size circles will make the shape of the cloud more natural.
2. Cut the circles in half and also overlap them.
We did it differently: Polya wanted to fold the circles in half and glue only one half of the circle. We have already made other crafts this way and she liked this option.
When the paper is completely dry, you can finish drawing the sun's rays and flowers on the grass with a pencil. You can do this with plasticine. Let the baby choose for himself.
Mathematics lesson in 1st grade with the State Educational Institution on the topic: “Geometric figure: circle”
Purpose: To introduce the geometric figure - the circle. Learn to distinguish a circle from other geometric shapes and name it correctly. Fix the names of the colors. Cultivate respect for each other.
I Organizational moment.
1. Who goes to visit in the morning,
He acts wisely!
Taram-param, taram-param,
That's why it's morning!
Children, what time of day is it now? (morning)
After the morning comes... (day)
Often guests return when it comes... (evening) (With the help of pictures)
2. Look carefully at the pictures, what do they have in common? How are they all similar? (all pictures show the sun)
II. Subject message.
The sun is round. Today in the lesson we will get acquainted with a geometric figure - a circle. Let's learn to distinguish it from other figures, we will find round objects.
III. Getting to know the figure.
1. A guest came to our lesson - Winnie the Pooh. He arrived in hot air balloons. (Balloons are given to the children) The ball is round. (Offer to circle the ball with your palm or finger.)
2. Look at Winnie the Pooh, which parts of his body are round?
3. Winnie the Pooh loves to eat, and therefore brought with him a set of dishes (plane images of round and square dishes). But Winnie the Pooh likes to eat only from round dishes. Help me choose round dishes.
4. While Winnie the Pooh was getting to us, several plates broke. Help, glue them together! (Children collect a cut picture)
What shape is the plate?
5. Look around, find round objects in our class.
IV. Phys. a minute (round dance)
In an even circle one after another
We are going step by step.
Together everything is in place
Let's do it like this!
(The driver is selected one by one)
V. Consolidation of what has been learned
1. Winnie the Pooh has many friends. He brought their portraits. (Images of geometric shapes. We look at it and discuss who it is).
Tell me, what is round?
2. Children are given sets of geometric figures. Find a circle. (Tactile examination, roll a circle on the table). Discuss the color and size of the shapes.
Why does the circle roll? (because there are no corners)
Why are the wheels round? (because there are no corners, they can roll)
3. Laying out a sample image from the geom set. figures. (Vinny's friend)
VI. Work in a notebook.
- Finger gymnastics.
- Explanation of the task.
- Work in a notebook.
VII. Result: What figure did you meet? What did you do in class?
