Polygons. Visual Guide (2019). Regular polygon. Number of sides of a regular polygon

In this lesson we will begin a new topic and introduce a new concept for us: “polygon”. We will look at the basic concepts associated with polygons: sides, vertex angles, convexity and nonconvexity. Then we will prove the most important facts, such as the theorem on the sum of the internal angles of a polygon, the theorem on the sum of the external angles of a polygon. As a result, we will come close to studying special cases of polygons, which will be considered in further lessons.

Topic: Quadrilaterals

Lesson: Polygons

In the geometry course, we study the properties of geometric figures and have already examined the simplest of them: triangles and circles. At the same time, we also discussed specific special cases of these figures, such as right, isosceles and regular triangles. Now it's time to talk about more general and complex figures - polygons.

With a special case polygons we are already familiar - this is a triangle (see Fig. 1).

Rice. 1. Triangle

The name itself already emphasizes that this is a figure with three angles. Therefore, in polygon there can be many of them, i.e. more than three. For example, let’s draw a pentagon (see Fig. 2), i.e. figure with five corners.

Rice. 2. Pentagon. Convex polygon

Definition.Polygon- a figure consisting of several points (more than two) and the corresponding number of segments that sequentially connect them. These points are called peaks polygon, and the segments are parties. In this case, no two adjacent sides lie on the same straight line and no two non-adjacent sides intersect.

Definition.Regular polygon is a convex polygon in which all sides and angles are equal.

Any polygon divides the plane into two areas: internal and external. The internal area is also referred to as polygon.

In other words, for example, when they talk about a pentagon, they mean both its entire internal region and its border. And the internal region includes all points that lie inside the polygon, i.e. the point also refers to the pentagon (see Fig. 2).

Polygons are also sometimes called n-gons to emphasize that the general case of the presence of some unknown number of angles (n pieces) is considered.

Definition. Polygon perimeter- the sum of the lengths of the sides of the polygon.

Now we need to get acquainted with the types of polygons. They are divided into convex And non-convex. For example, the polygon shown in Fig. 2 is convex, and in Fig. 3 non-convex.

Rice. 3. Non-convex polygon

Definition 1. Polygon called convex, if when drawing a straight line through any of its sides, the entire polygon lies only on one side of this straight line. Non-convex are everyone else polygons.

It is easy to imagine that when extending any side of the pentagon in Fig. 2 it will all be on one side of this straight line, i.e. it is convex. But when drawing a straight line through a quadrilateral in Fig. 3 we already see that it divides it into two parts, i.e. it is not convex.

But there is another definition of the convexity of a polygon.

Definition 2. Polygon called convex, if when choosing any two of its interior points and connecting them with a segment, all points of the segment are also interior points of the polygon.

A demonstration of the use of this definition can be seen in the example of constructing segments in Fig. 2 and 3.

Definition. Diagonal of a polygon is any segment connecting two non-adjacent vertices.

To describe the properties of polygons, there are two most important theorems about their angles: theorem on the sum of interior angles of a convex polygon And theorem on the sum of exterior angles of a convex polygon. Let's look at them.

Theorem. On the sum of interior angles of a convex polygon (n-gon).

Where is the number of its angles (sides).

Proof 1. Let us depict in Fig. 4 convex n-gon.

Rice. 4. Convex n-gon

From the vertex we draw all possible diagonals. They divide an n-gon into triangles, because each of the sides of the polygon forms a triangle, except for the sides adjacent to the vertex. It is easy to see from the figure that the sum of the angles of all these triangles will be exactly equal to the sum of the internal angles of the n-gon. Since the sum of the angles of any triangle is , then the sum of the internal angles of an n-gon is:

Q.E.D.

Proof 2. Another proof of this theorem is possible. Let's draw a similar n-gon in Fig. 5 and connect any of its interior points with all vertices.

Rice. 5.

We have obtained a partition of the n-gon into n triangles (as many sides as there are triangles). The sum of all their angles is equal to the sum of the interior angles of the polygon and the sum of the angles at the interior point, and this is the angle. We have:

Q.E.D.

Proven.

According to the proven theorem, it is clear that the sum of the angles of an n-gon depends on the number of its sides (on n). For example, in a triangle, and the sum of the angles is . In a quadrilateral, and the sum of the angles is, etc.

Theorem. On the sum of external angles of a convex polygon (n-gon).

Where is the number of its angles (sides), and , …, are the external angles.

Proof. Let us depict a convex n-gon in Fig. 6 and designate its internal and external angles.

Rice. 6. Convex n-gon with designated external angles

Because The external angle is connected to the internal one as adjacent, and the same is true for the remaining external angles. Then:

During the transformations, we used the already proven theorem about the sum of internal angles of an n-gon.

Proven.

An interesting fact follows from the proven theorem that the sum of the external angles of a convex n-gon is equal to the number of its angles (sides). By the way, in contrast to the sum of internal angles.

Bibliography

1. Alexandrov A.D. and others. Geometry, 8th grade. - M.: Education, 2006.
2. Butuzov V.F., Kadomtsev S.B., Prasolov V.V. Geometry, 8th grade. - M.: Education, 2011.
3. Merzlyak A.G., Polonsky V.B., Yakir S.M. Geometry, 8th grade. - M.: VENTANA-GRAF, 2009.

1. Profmeter.com.ua ().
2. Narod.ru ().
3. Xvatit.com ().

Homework

In the course of geo-metry, we study the properties of geo-metric figures and have already looked at the simplest of them: triangles and surroundings. At the same time, we also discussed specific special cases of these figures, such as rectangular, equal and right tri-coal-ni-ki. Now the time has come to talk about more general and complex figures - a lot of coal.

With a private case a lot of coal we already know - this is a triangle (see Fig. 1).

Rice. 1. Triangle

In the name itself, it already underscores that this is a fi-gu-ra, which has three corners. Next, in a lot of coal there can be many of them, i.e. more than three. For example, draw a pentagon (see Fig. 2), i.e. fi-gu-ru with five corners-la-mi.

Rice. 2. Penta-corner. You-bumpy polygon

Definition.Polygon- figure, consisting of several points (more than two) and corresponding to the number of points from the kov, who follow them together. These points are called top-she-on-mi a lot of coal, but from cutting - hundred-ro-na-mi. In this case, no two adjacent sides lie on the same straight line and no two non-adjacent sides intersect .

Definition.Right polygon- this is a convex polygon, which has all the sides and angles equal.

Any polygon divides the plane into two areas: internal and external. The internal area is also from a lot of coal.

In other words, for example, when they talk about the pentagon, they mean both its entire internal region and its boundaries. tsu. And all the points that lie inside a lot of coal are related to the inner region, i.e. the point is also from-no-sit-xia to the five-coal-ni-ku (see Fig. 2).

A lot of coal is sometimes called n-coal to emphasize that it is common the case of an unknown number of angles (n pieces).

Definition. Peri-meter of many-coal-no-ka- the sum of the lengths of the sides of a lot of coal.

Now we need to get acquainted with the sights of a lot of coals. They are divided into you fart And farts. For example, the polygon shown in Fig. 2, you appear to be farting, and in Fig. 3 not fart.

Rice. 3. Nevy-bumpy polygon

2. Convex and non-convex polygons

Definition 1. Polygon na-za-va-et-sya you fart, if, when passing directly through any of its sides, the entire polygon lies only on one side of this straight line. Neva-puk-ly-mi everyone else appears a lot of coal.

It is easy to imagine that when extending any side of the five-corner in Fig. 2 all of it will turn out to be one side away from this straight line, i.e. he's farty. But when passing straight through the four-coal in Fig. 3 we already see that she divides it into two parts, i.e. he's not a fart.

But there is another definition of how much coal you have.

Definition 2. Polygon na-za-va-et-sya you fart, if when you select any two of its internal points and when connecting them from a cut, all points from the cut are also internal - not exactly a lot of coal.

A demonstration of the use of this definition can be seen in the example of the construction of cut-offs in Fig. 2 and 3.

Definition. Dia-go-na-lew a lot of coal is called any cut that connects two non-adjacent tops of it.

3. Theorem on the sum of interior angles of a convex n-gon

To describe the properties of polygons, there are two important theorems about their angles: theo-re-ma about the sum of the internal angles of a lot of angles And theo-re-ma about the sum of external angles of a lot of angles. Let's look at them.

Theorem. About the sum of the internal angles you have a lot of angles (n-coal-no-ka).

Where is the number of its angles (sides).

Proof 1. Illustration in Fig. 4 protruding n-gon.

Rice. 4. You-bumpy n-gon

From the top we will conduct all possible dia-gos. They divide n-gon-nik into tri-gon-nik, because. Each of the sides forms a lot of coal, except for the sides lying towards the top. It is easy to see from the figure that the sum of the angles of all these triangles will be exactly equal to the sum of the internal angles of the n-corner. Since the sum of the angles of any triangle is , then the sum of the internal angles of an n-angle:

Reason 2. It is possible that there is another reason for this theorem. Illustration of an analogous n-gon in Fig. 5 and connect any of its internal points with all the vertices.

We have divided the n-coal into n triangles (how many sides, so many triangles) ). The sum of all their angles is equal to the sum of the internal angles of the polygon and the sum of the angles at the internal point, and this is the angle. We have:

Q.E.D.

Do-ka-za-but.

According to the previous theory, it is clear that the sum of the angles n-coal does not depend on the number of its sides (from n). For example, in a triangle, the sum of the angles is . In wh-you-re-re-coal-no-ke, and the sum of the angles - etc.

4. Theorem on the sum of external angles of a convex n-gon

Theorem. About the sum of the external angles of a lot of coal (n-coal-no-ka).

Where is the number of its angles (sides), and , ..., are the external angles.

Proof. The image of a convex n-gon in Fig. 6 and designate its internal and external angles.

Rice. 6. You-convex n-gon with designated external corners

Because The external angle is connected with the internal angle as adjacent, and then analogous for the other external angles. Then:

During the pre-development, we have already used the theorem about the sum of internal angles n-coal-ni- ka.

Do-ka-za-but.

From the previous theorem it follows an interesting fact that the sum of the external angles of the convex n-coal is equal to -what are its angles (sides). By the way, depending on the sum of the internal angles.

Next, we will work in more detail with the particular case of a lot of coal - why-you-re-re-coal-no-mi. In the next lesson, we will get to know such a figure as par-ral-le-lo-gram, and discuss its properties.

SOURCE

http://interneturok.ru/ru/school/geometry/8-klass/chyotyrehugolniki/mnogougolniki

http://interneturok.ru/ru/school/geometry/8-klass/povtorenie/pryamougolnye-treugolniki

http://interneturok.ru/ru/school/geometry/8-klass/povtorenie/treugolniki-2

http://nsportal.ru/shkola/geometriya/library/2013/10/10/mnogougolniki-urok-v-8-klasse

https://im0-tub-ru.yandex.net/i?id=daa2ea7bbc3c92be3a29b22d8106e486&n=33&h=190&w=144

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Types of polygons:

Quadrilaterals

Quadrilaterals, respectively, consist of 4 sides and angles.

Sides and angles opposite each other are called opposite.

Diagonals divide convex quadrilaterals into triangles (see picture).

The sum of the angles of a convex quadrilateral is 360° (using the formula: (4-2)*180°).

Parallelograms

Parallelogram is a convex quadrilateral with opposite parallel sides (number 1 in the figure).

Opposite sides and angles in a parallelogram are always equal.

And the diagonals at the intersection point are divided in half.

Trapeze

Trapezoid- this is also a quadrilateral, and in trapezoids Only two sides are parallel, which are called reasons. Other sides are sides.

The trapezoid in the figure is numbered 2 and 7.

As in a triangle:

If the sides are equal, then the trapezoid is isosceles;

If one of the angles is right, then the trapezoid is rectangular.

The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.

Rhombus

Rhombus is a parallelogram in which all sides are equal.

In addition to the properties of a parallelogram, rhombuses have their own special property - The diagonals of a rhombus are perpendicular each other and bisect the corners of a rhombus.

In the picture there is a rhombus number 5.

Rectangles

Rectangle is a parallelogram in which each angle is right (see figure number 8).

In addition to the properties of a parallelogram, rectangles have their own special property - The diagonals of the rectangle are equal.

Squares

Square is a rectangle with all sides equal (No. 4).

It has the properties of a rectangle and a rhombus (since all sides are equal).

Subject, student age: geometry, 9th grade

Purpose of the lesson: study types of polygons.

Educational task: to update, expand and generalize students’ knowledge about polygons; form an idea of ​​the “component parts” of a polygon; conduct a study of the number of constituent elements of regular polygons (from triangle to n-gon);

Developmental task: to develop the ability to analyze, compare, draw conclusions, develop computational skills, oral and written mathematical speech, memory, as well as independence in thinking and learning activities, the ability to work in pairs and groups; develop research and educational activities;

Educational task: to cultivate independence, activity, responsibility for the assigned work, perseverance in achieving the goal.

During the classes: quote written on the board

“Nature speaks the language of mathematics, the letters of this language ... mathematical figures.” G.Galliley

At the beginning of the lesson, the class is divided into working groups (in our case, divided into groups of 4 people each - the number of group members is equal to the number of question groups).

1.Call stage-

Goals:

a) updating students’ knowledge on the topic;

b) awakening interest in the topic being studied, motivating each student for educational activities.

Technique: Game “Do you believe that...”, organization of work with text.

Forms of work: frontal, group.

“Do you believe that...”

1. ... the word “polygon” indicates that all the figures in this family have “many angles”?

2. ... does a triangle belong to a large family of polygons, distinguished among many different geometric shapes on a plane?

3. ... is a square a regular octagon (four sides + four corners)?

Today in the lesson we will talk about polygons. We learn that this figure is limited by a closed broken line, which in turn can be simple, closed. Let's talk about the fact that polygons can be flat, regular, or convex. One of the flat polygons is a triangle, with which you have long been familiar (you can show students posters depicting polygons, a broken line, show their different types, you can also use TSO).

2. Conception stage

Goal: obtaining new information, understanding it, selecting it.

Technique: zigzag.

Forms of work: individual->pair->group.

Each member of the group is given a text on the topic of the lesson, and the text is compiled in such a way that it includes both information already known to the students and information that is completely new. Along with the text, students receive questions, the answers to which must be found in this text.

Polygons. Types of polygons.

Who hasn't heard about the mysterious Bermuda Triangle, in which ships and planes disappear without a trace? But the triangle, familiar to us from childhood, is fraught with a lot of interesting and mysterious things.

In addition to the types of triangles already known to us, divided by sides (scalene, isosceles, equilateral) and angles (acute, obtuse, rectangular), the triangle belongs to a large family of polygons, distinguished among many different geometric shapes on the plane.

The word “polygon” indicates that all figures in this family have “many angles.” But this is not enough to characterize the figure.

A broken line A 1 A 2 ...A n is a figure that consists of points A 1, A 2, ...A n and the segments connecting them A 1 A 2, A 2 A 3,.... The points are called the vertices of the polyline, and the segments are called the links of the polyline. (Fig.1)

A broken line is called simple if it has no self-intersections (Fig. 2, 3).

A polyline is called closed if its ends coincide. The length of a broken line is the sum of the lengths of its links (Fig. 4).

A simple closed broken line is called a polygon if its neighboring links do not lie on the same straight line (Fig. 5).

Substitute a specific number, for example 3, in the word “polygon” instead of the “many” part. You will get a triangle. Or 5. Then - a pentagon. Note that, as many angles as there are, there are as many sides, so these figures could well be called polylaterals.

The vertices of the broken line are called the vertices of the polygon, and the links of the broken line are called the sides of the polygon.

The polygon divides the plane into two areas: internal and external (Fig. 6).

A plane polygon or polygonal area is the finite part of a plane bounded by a polygon.

Two vertices of a polygon that are the ends of one side are called adjacent. Vertices that are not ends of one side are non-neighboring.

A polygon with n vertices, and therefore n sides, is called an n-gon.

Although the smallest number of sides of a polygon is 3. But triangles, when connected to each other, can form other figures, which in turn are also polygons.

Segments connecting non-adjacent vertices of a polygon are called diagonals.

A polygon is called convex if it lies in the same half-plane relative to any line containing its side. In this case, the straight line itself is considered to belong to the half-plane.

The angle of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex.

Let's prove the theorem (about the sum of the angles of a convex n-gon): The sum of the angles of a convex n-gon is equal to 180 0 *(n - 2).

Proof. In the case n=3 the theorem is valid. Let A 1 A 2 ...A n be a given convex polygon and n>3. Let's draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals divide it into n – 2 triangles. The sum of the angles of a polygon is the sum of the angles of all these triangles. The sum of the angles of each triangle is equal to 180 0, and the number of these triangles n is 2. Therefore, the sum of the angles of a convex n-gon A 1 A 2 ...A n is equal to 180 0 * (n - 2). The theorem has been proven.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at this vertex.

A convex polygon is called regular if all its sides are equal and all angles are equal.

So the square can be called differently - a regular quadrilateral. Equilateral triangles are also regular. Such figures have long been of interest to craftsmen who decorated buildings. They made beautiful patterns, for example on parquet. But not all regular polygons could be used to make parquet. Parquet cannot be made from regular octagons. The fact is that each angle is equal to 135 0. And if some point is the vertex of two such octagons, then they will account for 270 0, and there is no place for the third octagon to fit there: 360 0 - 270 0 = 90 0. But for a square this is enough. Therefore, you can make parquet from regular octagons and squares.

The stars are also correct. Our five-pointed star is a regular pentagonal star. And if you rotate the square around the center by 45 0, you get a regular octagonal star.

1 group

What is a broken line? Explain what vertices and links of a polyline are.

Which broken line is called simple?

Which broken line is called closed?

What is a polygon called? What are the vertices of a polygon called? What are the sides of a polygon called?

2nd group

Which polygon is called flat? Give examples of polygons.

What is n – square?

Explain which vertices of a polygon are adjacent and which are not.

What is the diagonal of a polygon?

3 group

Which polygon is called convex?

Explain which angles of a polygon are external and which are internal?

Which polygon is called regular? Give examples of regular polygons.

4 group

What is the sum of the angles of a convex n-gon? Prove it.

Students work with the text, look for answers to the questions posed, after which expert groups are formed, in which work is carried out on the same issues: students highlight the main points, draw up a supporting summary, and present information in one of the graphic forms. Upon completion of work, students return to their work groups.

3. Reflection stage -

a) assessment of one’s knowledge, challenge to the next step of knowledge;

b) comprehension and appropriation of the information received.

Reception: research work.

Forms of work: individual->pair->group.

Working groups include specialists in answering each section of the proposed questions.

Returning to the working group, the expert introduces the answers to his questions to other group members. The group exchanges information between all members of the working group. Thus, in each working group, thanks to the work of experts, a general understanding of the topic being studied is formed.

Students' research work - filling out the table.

Solving interesting problems on the topic of the lesson.

• In a quadrilateral, draw a straight line so that it divides it into three triangles.
• How many sides does a regular polygon have, each of its interior angles measuring 135 0?
• In a certain polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be equal to: 360 0, 380 0?

Summing up the lesson. Recording homework.