Polygon made from triangles. Regular polygon
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What is called a polygon? Types of polygons. POLYGON, a planar geometric figure with three or more sides intersecting at three or more points (vertices). Definition. A polygon is a geometric figure bounded on all sides by a closed polyline, consisting of three or more segments (links). A triangle is definitely a polygon. A polygon is a shape with five or more corners.
Definition. A quadrilateral is a flat geometric figure consisting of four points (vertices of a quadrilateral) and four consecutive line segments (sides of a quadrilateral).
A rectangle is a rectangle with all its corners straight. They are named according to the number of sides or vertices: TRIANGLE (three-sided); FOUR-SIDE (four-sided); PENTAGON (five-sided), etc. In elementary geometry, M. is a figure bounded by straight lines, called sides. The points at which the sides intersect are called vertices. A polygon has more than three corners. So it is accepted or agreed.
Triangle - it is a triangle. And the quadrangle is also not a polygon, and it is not called a quadrangle either - it is either a square, or a rhombus, or a trapezoid. The fact that a polygon with three sides and three corners has its own name "triangle" does not deprive it of its polygon status.
See what "POLYGON" is in other dictionaries:
We learn that this figure is bounded by a closed polyline, which in turn is simple, closed. Let's talk about the fact that polygons are flat, regular, convex. Who has not heard of 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.
Although, of course, a figure consisting of three corners can also be considered a polygon
But this is not enough to characterize the figure. A broken line A1A2 ... An is a figure that consists of points A1, A2, ... An and the segments A1A2, A2A3, ... connecting them. A simple closed broken line is called a polygon if its adjacent links do not lie on one straight line (Fig. 5). Substitute a specific number in the word “polygon” instead of the part “many”, for example 3. You will get a triangle. Note that there are as many sides as there are angles, so these figures could well be called multilaterals.
Let A1A2 ... And n be a given convex polygon and n> 3. Draw in it (from one vertex) diagonals
The sum of the angles of each triangle is 1800, and the number of these triangles is n - 2. Therefore, the sum of the angles of the convex n - gon A1A2 ... And n is 1800 * (n - 2). The theorem is proved. The outer angle of a convex polygon at a given vertex is the angle adjacent to the inner corner of the polygon at this vertex.
In a quadrilateral, draw a line so that it divides it into three triangles
A quadrilateral never has three vertices on one straight line. The word “polygon” indicates that all shapes in this family have “many angles”. A broken line is called simple if it does not have self-intersections (Fig. 2, 3).
The length of a broken line is the sum of the lengths of its links (Fig. 4). In the case n = 3, the theorem is valid. So the square can be called in another way - a regular quadrangle. Such figures have long been of interest to masters who decorate buildings.
The number of vertices is equal to the number of sides. A broken line is called closed if its ends coincide. They made beautiful patterns, for example on the parquet. Our five pointed star Is a regular pentagonal star.
But not all regular polygons could be folded into parquet. Let's take a closer look at two types of polygons: a triangle and a quadrilateral. A polygon in which all interior angles are equal is called regular. Polygons are named according to the number of sides or vertices in it.
The part of the plane bounded by a closed polyline is called a polygon.
The segments of this polyline are called parties polygon. AB, BC, CD, DE, EA (Fig. 1) - sides of the polygon ABCDE. The sum of all sides of a polygon is called it perimeter.
The polygon is called convex, if it is located on one side of any of its sides, extended indefinitely beyond both vertices.
Polygon MNPKO (Fig. 1) will not be convex, since it is not located on one side of the straight line KP.
We will only consider convex polygons.
The angles made up of two adjacent sides of a polygon are called its internal corners, and their tops - the vertices of the polygon.
A straight line segment connecting two non-adjacent vertices of a polygon is called the diagonal of the polygon.
AC, AD - the diagonals of the polygon (Fig. 2).
The corners adjacent to the inner corners of the polygon are called the outer corners of the polygon (Figure 3).
Depending on the number of angles (sides), the polygon is called a triangle, quadrilateral, pentagon, etc.
Two polygons are said to be equal if they can be overlapped.
Inscribed and circumscribed polygons
If all vertices of a polygon lie on a circle, then the polygon is called inscribed into a circle, and a circle - described near the polygon (fig).
If all sides of the polygon are tangent to the circle, then the polygon is called described about a circle, and the circle is called inscribed into a polygon (fig).
Similarity of polygons
Two polygons of the same name are called similar if the angles of one of them are respectively equal to the angles of the other, and the similar sides of the polygons are proportional.
Polygons with the same name are called polygons that have the same number of sides (angles).
The sides of such polygons connecting the vertices of correspondingly equal angles are called similar (Fig).
So, for example, for the polygon ABCDE to be similar to the polygon A'B'C'D'E ', it is necessary that: ∠A = ∠A' ∠B = ∠B '∠С = ∠С' ∠D = ∠D '∠ E = ∠E 'and, in addition, AB / A'B' = BC / B'C '= CD / C'D' = DE / D'E '= EA / E'A'.
The ratio of the perimeters of similar polygons
First, consider the property of a series of equal relationships. Let us have, for example, the ratios: 2/1 = 4/2 = 6/3 = 8/4 = 2.
Let's find the sum of the previous members of these relations, then - the sum of their subsequent members and find the ratio of the sums received, we get:
$$ \ frac (2 + 4 + 6 + 8) (1 + 2 + 3 + 4) = \ frac (20) (10) = 2 $$
We get the same if we take a number of some other relations, for example: 2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 2/3 Let us find the sum of the previous terms of these relations and the sum of the subsequent ones, and then we find the ratio of these sums, we get:
$$ \ frac (2 + 4 + 5 + 8 + 10) (3 + 6 + 9 + 12 + 15) = \ frac (30) (45) = \ frac (2) (3) $$
In either case, the sum of the previous members of a series of equal relations refers to the sum of subsequent members of the same series, as the previous member of any of these relations refers to its subsequent one.
We have deduced this property by considering the series numerical examples... It can be deduced strictly and in general terms.
Now let's consider the ratio of the perimeters of such polygons.
Let the polygon ABCDE be similar to the polygon A'B'C'D'E '(fig).
From the similarity of these polygons it follows that
AB / A'B '= BC / B'C' = CD / C'D '= DE / D'E' = EA / E'A '
Based on the property we have derived for a number of equal relations, we can write:
The sum of the previous members of the relations we have taken is the perimeter of the first polygon (P), and the sum of the subsequent members of these relations is the perimeter of the second polygon (P ’), which means that P / P’ = AB / A’B ’.
Hence, the perimeters of such polygons are referred to as similar sides.
The ratio of the areas of similar polygons
Let ABCDE and A'B'C'D'E 'be similar polygons (fig).
It is known that ΔABC ~ ΔA'B'C 'ΔACD ~ ΔA'C'D' and ΔADE ~ ΔA'D'E '.
Besides,
;
Since the second ratios of these proportions are equal, which follows from the similarity of polygons, then
Using the property of a number of equal relations, we get:
Or 
where S and S 'are the areas of these similar polygons.
Hence, the areas of similar polygons are referred to as squares of similar sides.
The resulting formula can be converted to this form: S / S '= (AB / A'B') 2
Free polygon area
Let it be required to calculate the area of an arbitrary quadrilateral ABDC (fig).
Let's draw a diagonal in it, for example AD. We get two triangles ABD and ACD, the areas of which we know how to calculate. Then we find the sum of the areas of these triangles. The resulting sum will express the area of this quadrilateral.
If you need to calculate the area of a pentagon, then we do the same: draw diagonals from one of the vertices. We get three triangles, the areas of which we can calculate. This means that we can also find the area of the given pentagon. We do the same when calculating the area of any polygon.
Polygon projection area
Recall that the angle between a straight line and a plane is the angle between a given straight line and its projection onto a plane (Fig.).
Theorem. The area of the orthogonal projection of the polygon onto the plane is equal to the area of the projected polygon multiplied by the cosine of the angle formed by the plane of the polygon and the plane of the projection.
Each polygon can be divided into triangles, the sum of the areas of which is equal to the area of the polygon. Therefore, it suffices to prove the theorem for a triangle.
Let ΔABS be projected onto the plane R... Consider two cases:
a) one of the sides of ΔABS is parallel to the plane R;
b) none of the sides of ΔABS is parallel R.
Consider first case: let [AB] || R.
Let's draw a plane through (AB) R 1 || R and design orthogonally ΔABS on R 1 and on R(rice.); we get ΔABS 1 and ΔA'B'S '.
By the property of the projection, we have ΔABS 1 (cong) ΔА'В'С', and therefore
S Δ ABC1 = S Δ A'B'C '
Draw ⊥ and the segment D 1 C 1. Then ⊥, a \ (\ overbrace (CD_1C_1) \) = φ is the value of the angle between the plane ΔABS and the plane R 1 . That's why
S Δ ABC1 = 1/2 | AB | | C 1 D 1 | = 1/2 | AB | | CD 1 | cos φ = S Δ ABC cos φ
and therefore S Δ A'B'C '= S Δ ABC cos φ.
Let's move on to consideration second case... Let's draw a plane R 1 || R through that vertex ΔABS, the distance from which to the plane R the smallest (let it be vertex A).
Let's design ΔABS on a plane R 1 and R(rice.); let its projections be respectively ΔАВ 1 С 1 and ΔА'В'С '.
Let (ВС) ∩ p 1 = D. Then
S Δ A'B'C '= S ΔAB1 C1 = S ΔADC1 - S ΔADB1 = (S ΔADC - S ΔADB) cos φ = S Δ ABC cos φ
Other materialsSubject, student age: geometry, grade 9
The purpose of the lesson: the study of the types of polygons.
Learning task: to update, expand and generalize students' knowledge about polygons; to form an idea of the "constituent parts" of the polygon; conduct research on the number constituent elements regular polygons (from triangle to n - gon);
Developing 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; to develop research and cognitive activities;
Educational task: to educate independence, activity, responsibility for the assigned work, perseverance in achieving the set goal.
During the classes: there is a quote on the blackboard
"Nature speaks in 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, division into groups of 4 people in 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 under study, motivating each student for educational activities.
Technique: The game “Do you believe that ...”, the organization of work with the text.
Forms of work: frontal, group.
"Do you believe that ...."
1.… the word “polygon” indicates that all shapes in this family have “many angles”?
2. ... a triangle belongs to a large family of polygons, distinguished among many different geometric shapes on surface?
3.… is a square a regular octagon (four sides + four corners)?
Today's lesson will focus on polygons. We learn that this figure is bounded by a closed polyline, which in turn is simple, closed. Let's talk about the fact that polygons are flat, regular, convex. One of the flat polygons is a triangle, with which you have been familiar for a long time (you can demonstrate to students posters with the image of polygons, a broken line, show them different kinds, you can also use TCO).
2. Stage of comprehension
Purpose: receiving new information, its comprehension, selection.
Reception: zigzag.
Forms of work: individual-> pair-> group.
Each of the group is given a text on the topic of the lesson, and the text is composed in such a way that it includes both information already known to students and completely new information. Together with the text, students receive questions, the answers to which must be found in this text.
Polygons. Types of polygons.
Who has not heard of 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.
In addition to the types of triangles already known to us, divided by sides (versatile, isosceles, equilateral) and corners (acute-angled, obtuse, right-angled), a triangle belongs to a large family of polygons, distinguished among many different geometric shapes on the plane.
The word “polygon” indicates that all shapes in this family have “many angles”. But this is not enough to characterize the figure.
A broken line А 1 А 2 ... А n is a figure that consists of points А 1, А 2, ... А n and the segments А 1 А 2, А 2 А 3, ... connecting them. 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 does not have self-intersections (Fig. 2, 3).
A broken line 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 adjacent links do not lie on one straight line (Fig. 5).
Substitute a specific number in the word “polygon” instead of the part “many”, for example 3. You will get a triangle. Or 5. Then - a pentagon. Note that there are as many sides as there are angles, so these figures could well be called multilaterals.
The vertices of the polyline are called the vertices of the polygon, and the links of the polyline are called the sides of the polygon.
The polygon divides the plane into two areas: internal and external (Fig. 6).
A flat polygon or polygonal region is the end portion 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 the ends of one side are not adjacent.
A polygon with n vertices, and hence with n sides, is called an n-gon.
Although the smallest number of sides of a polygon is 3. But triangles, connecting with each other, can form other shapes, which in turn are also polygons.
Line segments connecting non-adjacent vertices of a polygon are called diagonals.
A polygon is called convex if it lies in one half-plane with respect to any line containing its side. In this case, the 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 us prove the theorem (on the sum of the angles of a convex n - gon): The sum of the angles of a convex n - gon is 180 0 * (n - 2).
Proof. In the case n = 3, the theorem is valid. Let А 1 А 2 ... А n be a given convex polygon and n> 3. Draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals split it into n - 2 triangles. The sum of the angles of a polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 180 0, and the number of these triangles is n - 2. Therefore, the sum of the angles of a convex n - gon А 1 А 2 ... А n is equal to 180 0 * (n - 2). The theorem is proved.
The outer angle of a convex polygon at a given vertex is the angle adjacent to the inner corner of the polygon at this vertex.
A convex polygon is called regular if all sides of it are equal and all angles are equal.
So the square can be called in another way - a regular quadrangle. Equilateral triangles are also regular. Such figures have long been of interest to masters who decorate buildings. They made beautiful patterns, for example, on the parquet. But not all regular polygons could be folded into parquet. Parquet cannot be folded from regular octagons. The fact is that each angle of them is equal to 135 0. And if any point is the vertex of two such octagons, then their share will be 270 0, and there is nowhere for the third octagon to fit there: 360 0 - 270 0 = 90 0. But this is enough for a square. Therefore, it is possible to fold the 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.
1st group
What is called a broken line? Explain what the vertices and links of a polyline are.
Which polyline is called simple?
Which polyline is called closed?
What is called a polygon? What are the vertices of a polygon? What are the sides of a polygon?
2nd group
Which polygon is called flat? Give examples of polygons.
What is n - gon?
Explain which vertices of the polygon are adjacent and which are not.
What is the diagonal of a polygon?
Group 3
Which polygon is called convex?
Explain which corners of the 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, looking for answers to the questions posed, after which expert groups are formed, the work in which is on the same issues: students highlight the main thing, make up a supporting summary, present information in one of the graphic forms. At the end of the work, students return to their work groups.
3. Stage of reflection -
a) assessment of their 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.
In the working groups, there are specialists in answering each of the sections of the proposed questions.
Returning to the working group, the expert introduces the other members of the group with the answers to his questions. In the group, information is exchanged between all members of the working group. Thus, in each working group, thanks to the work of experts, is general idea on the topic under study.
Research work of students - filling out the table.
| Regular polygons | Drawing | Number of sides | Number of vertices | Sum of all inside corners | Degree measure int. corner | Outside angle measure | Number of diagonals |
| A) triangle | |||||||
| B) quadrangle | |||||||
| C) fivewolnik | |||||||
| D) hexagon | |||||||
| E) n-gon |
Solving interesting problems on the topic of the lesson.
- In the quadrilateral, draw a line so that it divides it into three triangles.
- How many sides does a regular polygon have, each of inner corners which is 135 0?
- In some 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. Homework recording.
In the course of geo-met-ry, we will study the properties of geo-metric figures and have already considered the simplest of them: triangles-ni-ki and surroundings. At the same time, we discussed specific cases of these figures, such as rectangular, equal tre-coal-ni-ki. Now the time has come to talk about more general and complex figs - a lot of coal-no-kah.
In a particular case many-coal-nikov we already know-to-we are a triangle (see Fig. 1).
Rice. 1. Tre-charcoal-nick
In the name itself, it is already under-cher-ki-va-et-sya, that this is fi-gu-ra, which has three corners. Left-to-wa-tel-but, in a lot of coal-no-ke there can be many of them, i.e. more than three. For example, the image of a five-coal-nickname (see Fig. 2), ie fi-gu-ru with five corners-la-mi.
Rice. 2. Five-sided nickname. You-bunch-a-yogo-coal-nick
Definition.Polygon- fi-gu-ra, consisting of several points (more than two) and co-with-the-vet-u-u-u-l-a-th-a-th kov, some of them after-va-tel-but co-unite. These points are na-zy-va-ut-Xia ver-shi-na-mi a lot of coal-no, but from-cut - sto-ro-na-mi... At the same time, no two adjacent sides lie on the same straight line and no two non-adjacent sides do not cross ...
Definition.Correct multi-coal-nick- this is you-bunch-a-lot-coal-nick, at which-ro-go all sides and angles are equal.
Any polygon div-de-la-et the plane-bone into two regions: inner-ren-nu and outer-nu. The inner-ren-nyu area is also related to a lot of coal-no-ku.
In other words, for example, when they talk about five-coal-no-ke, they mean its entire inner-nyu region, and tsu. And all the points that lie inside a lot of coal, i.e. the point also refers to the five-coal-ni-ku (see Fig. 2).
A lot of coal-ni-ki still sometimes na-zy-va-yut n-coal-ni-ka-mi, to underline that ras-smat-ri-va-is-Xia common the case of an unknown number of corners (n pieces).
Definition. Per-ri-meter many-coal-no-ka- the sum of the lengths of the sides of a lot of coal.
Now it is necessary to know-to-know-to-go with we-yes-many-coal-nikov. They are on you-fart-ly and not-fart-ly... For example, the polygonal nickname depicted in Fig. 2 is bulging, and in Fig. 3 non-bunched.
Rice. 3. Neva-bunch-ly-go-coal-nick
2. Convex and non-convex polygons
Definition-de-le-tion 1. Polygon na-zy-wa-et-Xia you-bunch-lym if, when pro-ve-de-ny, straight through any of its sides all polygon lies only one side of this straight line. Neva-bunch-ly-mi all the rest mogo-coal-ni-ki.
It is easy to imagine that when any side of the five-coal in Fig. 2 he will all be one hundred percent from this straight line, i.e. he is a bunch. But with the pro-ve-de-ny of a straight line through in four-you-rekh-coal-no-ke in Fig. 3 we already see that she divides it into two parts, i.e. he is not-fart-ly.
But there is also another definition of you-bunch-of-a lot of coal-no-ka.
Definition-de-le-tion 2. Polygon na-zy-wa-et-Xia you-bunch-lym if, when you select any two of its inner points and when they are connected, all points from the cut are also inner -n-mi-dots-mi-a-go-coal-ni-ka.
Demonstration of the use of this definition can be seen on the example of the construction of cut-off points in Fig. 2 and 3.
Definition. Dia-go-na-liu a lot of coal-no-ka-zy-va-is-Xia any ot-zok, co-one-nya-yu-shi-two not contiguity of its vertices.
3. The theorem on the sum of the interior angles of a convex n-gon
For a description of the properties of many-coal-nikov, there are two important theo-re-we about their angles: theo-re-ma about the sum of the inner-corners of the vy-bunch-lo-th-many-coal-no-ka and theo-re-ma about the sum of the outer angles of the bunch... Examine them.
Theorem. About the sum of the inner angles of the bunch of many-coal-no-ka (n-coal-no-ka).
Where is the number of its corners (sides).
Do-ka-tel-tstvo 1. Image-zim in Fig. 4 you-bunch n-gon-nick.
Rice. 4. You-bunch-n-gon-nick
From the top-shi-we pro-we-dem all possible dia-go-na-li. They divide the n-gon-nick into the tri-gon-nick, because each of the sides has a lot of coal-no-ka-ra-zu-et a triangle, except for the sides that come to the top. It is easy to see from the drawing that the sum of the angles of all these triangles will be exactly equal to the sum of the inner angles of n-coals. Since the sum of the angles of any triangle is, then the sum of the inner angles of n-coal is:
Do-ka-tel-tstvo 2. Possibly, and another do-ka-tel-tiality of this theo-re-we. Pictured-ra-winter analogous n-gon in Fig. 5 and co-unite any of its internal points with all vertices.
We have-got-chi-whether raz-bi-e-n-coals-n-coals into n-coals (how many sides, so many coals ). The sum of all their angles is equal to the sum of the inner angles of the polygon and the sum of the angles at the inner point, and this is the angle. We have:
Q.E.D.
Do-ka-za-but.
According to the adjusted theory, it can be seen that the sum of the angles of n-coal-ni-ka-sits on the number of its sides (from n). For example, in a triangle, and the sum of the angles. In th-you-rekh-coal-no-ke, and the sum of the angles - etc.
4. The theorem on the sum of the outer angles of a convex n-gon
Theorem. About the sum of the outer angles of a bunch of a lot of coal-no-ka (n-coal-no-ka).
Where is the number of its corners (sides), and, ..., are the outer corners.
Proof. Pictured-ra-winter vy-bump-n-gon-nick in Fig. 6 and denote its inner and outer corners.
Rice. 6. You-bump-ly n-gon-nick with denoted outer corners
Because the outer corner is connected with the inner corner as adjacent, then 
and similarly for the rest of the outer corners. Then:
In the course of pre-ob-ra-zo-va-ny, we used-to-use-zo-va-li already reached the theorem about the sum of the inner angles n-coal-ni- ka.
Do-ka-za-but.
From the top-of-the-line theo-re-we follow the interesting fact that the sum of the outer angles of the bunch of n-coal is equal to 
from the number of its corners (sides). By the way, in contrast to the sum of the inner angles.
Further, we will work in more detail with the particular case of a lot of coal-nikov - che-you-rekh-coal-ni-ka-mi. In the next lesson, we will get to know such a fi-gu-swarm as pa-ra-le-lo-gram, and discuss its properties.
A 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
