Regular polygon. The number of sides of a regular polygon. Geometric figure polygon Polygon with three sides show
Knowledge of terminology, as well as knowledge of the properties of various geometric shapes, will help in solving many problems in geometry. Studying such a section as planimetry, the student often comes across the term “polygon”. What figure is characterized by this concept?
Polygon - definition of a geometric figure
A closed broken line, all sections of which lie in the same plane and do not have self-intersections, forms a geometric figure called a polygon. The number of links of the polyline must be at least 3. In other words, a polygon is defined as a part of a plane whose boundary is a closed broken line.
In the course of solving problems involving a polygon, concepts such as:
- The side of the polygon. This term characterizes a segment (link) of a broken chain of the desired figure.
- Polygon angle (internal) - the angle that is formed by 2 adjacent links of the polyline.
- The vertex of a polygon is defined as the vertex of a polyline.
- The diagonal of a polygon is a segment connecting any 2 vertices (except neighboring ones) of a polygonal figure.
In this case, the number of links and the number of vertices of the polyline within one polygon coincide. Depending on the number of corners (or segments of the broken line, respectively), the type of polygon is also determined:
- 3 corners - triangle.
- 4 corners - quadrilateral.
- 5 corners - pentagon, etc.
If a polygonal figure has equal angles and, accordingly, the sides, then they say that the given polygon is regular.
Types of polygons
All polygonal geometric shapes are divided into 2 types - convex and concave.
- If any of the sides of the polygon, after continuing to a straight line, does not form intersection points with the actual figure, then you have a convex polygonal figure.
- If, after the extension of the (any) side, the resulting line intersects the polygon, we are talking about a concave polygon.
Polygon Properties
Regardless of whether the studied polygonal figure is regular or not, it has the following properties. So:
- Its internal angles form (p – 2)*π in total, where
π is the radian measure of a straightened angle, corresponds to 180°,
p is the number of corners (vertices) of a polygonal figure (p-gon).
- The number of diagonals of any polygonal figure is determined from the ratio p * (p - 3) / 2, where
p is the number of sides of the p-gon.
Types of polygons:
Quadrangles
Quadrangles, respectively, consist of 4 sides and corners.
Sides and angles that are opposite each other are called opposite.
Diagonals divide convex quadrilaterals into triangles (see figure).
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 (numbered 1 in the figure).
Opposite sides and angles in a parallelogram are always equal.
And the diagonals at the point of intersection are divided in half.
Trapeze
Trapeze is also a quadrilateral, and trapeze only two sides are parallel, which are called grounds. The other sides are sides .
The trapezoid in the figure is numbered 2 and 7.
As in the 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 a trapezoid is half the sum of the bases and parallel to them.
Rhombus
Rhombus is a parallelogram with all sides 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 figure, the rhombus is numbered 5.
Rectangles
Rectangle- this is a parallelogram, in which each corner is a right one (see in the figure at 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 (#4).
It has the properties of a rectangle and a rhombus (since all sides are equal).
Topic: "Polygons. Types of polygons"
Grade 9
SL №20
Teacher: Kharitonovich T.I. The purpose of the lesson: the study of types of polygons.
Learning task: update, expand and generalize students' knowledge of polygons; form an idea of the “components” of a polygon; conduct a study of the number of constituent elements of regular polygons (from a triangle to n-gon);
Development task: 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 ability to work in pairs and groups; develop research and cognitive activity;
Educational task: to cultivate independence, activity, responsibility for the task assigned, perseverance in achieving the goal.
Equipment: interactive whiteboard (presentation)
During the classes
Show presentation: "Polygons"
“Nature speaks the language of mathematics, the letters of this language ... mathematical figures". G. Gallilei
At the beginning of the lesson, the class is divided into working groups (in our case, division into 3 groups)
1. Call stage-
a) updating students' knowledge on the topic;
b) the awakening of interest in the topic under study, the motivation of each student for learning activities.
Reception: The 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 of this family have "many corners"?
2. … does a triangle belong to a large family of polygons distinguished from a variety of 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 bounded by a closed broken line, which in turn can be simple, closed. Let's talk about the fact that polygons are flat, regular, convex. One of the flat polygons is a triangle that you have been familiar with for a long time (you can show students posters depicting polygons, a broken line, show their various types, you can also use TCO).
2. Stage of comprehension
Purpose: obtaining new information, its comprehension, selection.
Reception: zigzag.
Forms of work: individual->pair->group.
Each group is given a text on the topic of the lesson, and the text is designed 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 things.
In addition to the types of triangles already known to us, divided by sides (scalene, isosceles, equilateral) and angles (acute-angled, obtuse-angled, right-angled), the triangle belongs to a large family of polygons, distinguished among many different geometric shapes on the plane.
The word "polygon" indicates that all the figures of this family have "many corners". 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 segments A1A2, A2A3,… 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 the same straight line (Fig. 5).
Substitute in the word “polygon” instead of the “many” part a specific number, for example 3. You will get a triangle. Or 5. Then - a pentagon. Note that there are as many angles as there are sides, 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 regions: internal and external (Fig. 6).
A plane polygon or polygonal region is a finite part of a plane bounded by a polygon.
Two vertices of a polygon that are ends of the same side are called neighbors. Vertices that are not ends of one side are non-adjacent.
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, connecting with each other, can form other shapes, which in turn are also polygons.
Segments connecting non-neighboring 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 that vertex.
Let's prove the theorem (on the sum of angles of a convex n-gon): The sum of the angles of a convex n-gon is equal to 1800*(n - 2).
Proof. In the case n=3 the theorem is true. Let А1А2…А 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 the polygon is the same as the sum of the angles of all these triangles. 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 a convex n - angle A1A2 ... A n is 1800 * (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 that vertex.
A convex polygon is called regular if all 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 the masters who decorated the buildings. They made beautiful patterns, for example, on the parquet. But not all regular polygons could be used to form parquet. Parquet cannot be formed from regular octagons. The fact is that they have each angle equal to 1350. And if any point is the vertex of two such octagons, then they will have 2700, and there is nowhere for the third octagon to fit: 3600 - 2700 = 900. But for a square this is enough. Therefore, it is possible to fold the parquet from regular octagons and squares.
The stars are correct. Our five-pointed star is a regular pentagonal star. And if you rotate the square around the center by 450, you get a regular octagonal star.
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? What are the vertices of a polygon called? What are the sides of a polygon?
What is a flat polygon? 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?
What is a convex polygon?
Explain which corners of the polygon are external and which are internal?
What is a regular polygon? Give examples of regular polygons.
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 thing, draw up a supporting abstract, present information in one of the graphic forms. At the end of the work, students return to their working groups.
3. Stage of reflection -
a) assessment of their knowledge, challenge to the next step of knowledge;
b) understanding and appropriation of the received information.
Reception: research work.
Forms of work: individual->pair->group.
The working groups are experts in the answers to 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 their questions. In the group there is an exchange of information of all members of the working group. Thus, in each working group, thanks to the work of experts, it develops general idea on the topic under study.
Research students- filling in the table.
Regular polygons Drawing Number of sides Number of vertices Sum of all internal angles Degree measure of internal. angle Degree measure of external angle Number of diagonals
A) a triangle
B) quadrilateral
B) five-hole
D) hexagon
E) n-gon
Solving interesting problems on the topic of the lesson.
1) How many sides does a regular polygon have, each of whose internal angles is equal to 1350?
2) In a certain polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be: 3600, 3800?
3) Is it possible to build a pentagon with angles of 100,103,110,110,116 degrees?
Summing up the lesson.
Recording homework: STR 66-72 №15,17 AND PROBLEM: in a QUADRANGLE, DRAW A DIRECT SO THAT SHE DIVIDES IT INTO THREE TRIANGLES.
Reflection in the form of tests (on an interactive whiteboard)
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Triangle, square, hexagon - these figures are known to almost everyone. But not everyone knows what a regular polygon is. But this is all the same Regular polygon is called the one that has equal angles and sides. There are a lot of such figures, but they all have the same properties, and the same formulas apply to them.
Properties of regular polygons
Any regular polygon, be it a square or an octagon, can be inscribed in a circle. This basic property is often used when constructing a figure. In addition, a circle can also be inscribed in a polygon. In this case, the number of points of contact will be equal to the number of its sides. It is important that a circle inscribed in a regular polygon will have a common center with it. These geometric figures are subject to the same theorems. Any side of a regular n-gon is associated with the radius R of the circumscribed circle around it. Therefore, it can be calculated using the following formula: a = 2R ∙ sin180°. Through you can find not only the sides, but also the perimeter of the polygon.
How to find the number of sides of a regular polygon
Any one consists of a certain number of segments equal to each other, which, when connected, form a closed line. In this case, all the corners of the formed figure have the same value. Polygons are divided into simple and complex. The first group includes a triangle and a square. Complex polygons have more sides. They also include star-shaped figures. For complex regular polygons, the sides are found by inscribing them in a circle. Let's give a proof. Draw a regular polygon with an arbitrary number of sides n. Describe a circle around it. Specify the radius R. Now imagine that some n-gon is given. If the points of its angles lie on a circle and are equal to each other, then the sides can be found by the formula: a = 2R ∙ sinα: 2.
Finding the number of sides of an inscribed right triangle
An equilateral triangle is a regular polygon. The same formulas apply to it as to the square and the n-gon. A triangle will be considered correct if it has the same length sides. In this case, the angles are 60⁰. Construct a triangle with given side length a. Knowing its median and height, you can find the value of its sides. To do this, we will use the method of finding through the formula a \u003d x: cosα, where x is the median or height. Since all sides of the triangle are equal, we get a = b = c. Then the following statement is true: a = b = c = x: cosα. Similarly, you can find the value of the sides in an isosceles triangle, but x will be the given height. At the same time, it should be projected strictly on the base of the figure. So, knowing the height x, we find the side a of an isosceles triangle using the formula a \u003d b \u003d x: cosα. After finding the value of a, you can calculate the length of the base c. Let's apply the Pythagorean theorem. We will look for the value of half the base c: 2=√(x: cosα)^2 - (x^2) = √x^2 (1 - cos^2α) : cos^2α = x ∙ tgα. Then c = 2xtanα. In such a simple way, you can find the number of sides of any inscribed polygon.
Calculating the sides of a square inscribed in a circle
Like any other inscribed regular polygon, a square has equal sides and angles. The same formulas apply to it as to the triangle. You can calculate the sides of a square using the value of the diagonal. Let's consider this method in more detail. It is known that the diagonal bisects the angle. Initially, its value was 90 degrees. Thus, after division, two are formed. Their angles at the base will be equal to 45 degrees. Accordingly, each side of the square will be equal, that is: a \u003d b \u003d c \u003d d \u003d e ∙ cosα \u003d e √ 2: 2, where e is the diagonal of the square, or the base of the right triangle formed after division. This is not the only way to find the sides of a square. Let's inscribe this figure in a circle. Knowing the radius of this circle R, we find the side of the square. We will calculate it as follows a4 = R√2. The radii of regular polygons are calculated by the formula R \u003d a: 2tg (360 o: 2n), where a is the length of the side.
How to calculate the perimeter of an n-gon
The perimeter of an n-gon is the sum of all its sides. It is easy to calculate it. To do this, you need to know the values of all sides. For some types of polygons, there are special formulas. They allow you to find the perimeter much faster. It is known that any regular polygon has equal sides. Therefore, in order to calculate its perimeter, it is enough to know at least one of them. The formula will depend on the number of sides of the figure. In general, it looks like this: P \u003d an, where a is the value of the side, and n is the number of angles. For example, to find the perimeter of a regular octagon with a side of 3 cm, you need to multiply it by 8, that is, P = 3 ∙ 8 = 24 cm. For a hexagon with a side of 5 cm, we calculate as follows: P = 5 ∙ 6 = 30 cm. And so for each polygon.
Finding the perimeter of a parallelogram, square and rhombus
Depending on how many sides a regular polygon has, its perimeter is calculated. This makes the task much easier. Indeed, unlike other figures, in this case it is not necessary to look for all its sides, just one is enough. By the same principle, we find the perimeter of quadrangles, that is, a square and a rhombus. Despite the fact that these are different figures, the formula for them is the same P = 4a, where a is the side. Let's take an example. If the side of a rhombus or square is 6 cm, then we find the perimeter as follows: P \u003d 4 ∙ 6 \u003d 24 cm. A parallelogram has only opposite sides. Therefore, its perimeter is found using a different method. So, we need to know the length a and the width b of the figure. Then we apply the formula P \u003d (a + c) ∙ 2. A parallelogram, in which all sides and angles between them are equal, is called a rhombus.
Finding the perimeter of an equilateral and right triangle
The perimeter of the correct one can be found by the formula P \u003d 3a, where a is the length of the side. If it is unknown, it can be found through the median. IN right triangle only two sides are equal. The basis can be found through the Pythagorean theorem. After the values of all three sides become known, we calculate the perimeter. It can be found by applying the formula P \u003d a + b + c, where a and b are equal sides, and c is the base. Recall that in an isosceles triangle a \u003d b \u003d a, therefore, a + b \u003d 2a, then P \u003d 2a + c. For example, the side of an isosceles triangle is 4 cm, find its base and perimeter. We calculate the value of the hypotenuse according to the Pythagorean theorem c \u003d √a 2 + in 2 \u003d √16 + 16 \u003d √32 \u003d 5.65 cm. Now we calculate the perimeter P \u003d 2 ∙ 4 + 5.65 \u003d 13.65 cm.
How to find the angles of a regular polygon
A regular polygon occurs in our lives every day, for example, an ordinary square, triangle, octagon. It would seem that there is nothing easier than building this figure yourself. But this is just at first glance. In order to construct any n-gon, you need to know the value of its angles. But how do you find them? Even scientists of antiquity tried to build regular polygons. They guessed to fit them into circles. And then the necessary points were marked on it, connected by straight lines. For simple figures, the construction problem has been solved. Formulas and theorems have been obtained. For example, Euclid in his famous work "The Beginning" was engaged in solving problems for 3-, 4-, 5-, 6- and 15-gons. He found ways to construct them and find angles. Let's see how to do this for a 15-gon. First you need to calculate the sum of its internal angles. It is necessary to use the formula S = 180⁰(n-2). So, we are given a 15-gon, which means that the number n is 15. We substitute the data we know into the formula and get S = 180⁰ (15 - 2) = 180⁰ x 13 = 2340⁰. We have found the sum of all interior angles of a 15-gon. Now we need to get the value of each of them. There are 15 angles in total. We do the calculation of 2340⁰: 15 = 156⁰. This means that each internal angle is 156⁰, now using a ruler and a compass, you can build a regular 15-gon. But what about more complex n-gons? For centuries, scientists have struggled to solve this problem. It was only found in the 18th century by Carl Friedrich Gauss. He was able to build a 65537-gon. Since then, the problem has officially been considered completely solved.
Calculation of angles of n-gons in radians
Of course, there are several ways to find the corners of polygons. Most often they are calculated in degrees. But you can also express them in radians. How to do it? It is necessary to proceed as follows. First, we find out the number of sides of a regular polygon, then subtract 2 from it. So, we get the value: n - 2. Multiply the found difference by the number n ("pi" \u003d 3.14). Now it remains only to divide the resulting product by the number of angles in the n-gon. Consider these calculations using the example of the same fifteen-sided. So, the number n is 15. Let's apply the formula S = p(n - 2) : n = 3.14(15 - 2) : 15 = 3.14 ∙ 13: 15 = 2.72. This is of course not the only way to calculate an angle in radians. You can simply divide the size of the angle in degrees by the number 57.3. After all, that many degrees is equivalent to one radian.
Calculation of the value of angles in degrees
In addition to degrees and radians, you can try to find the value of the angles of a regular polygon in grads. This is done in the following way. Subtract 2 from the total number of angles, divide the resulting difference by the number of sides of a regular polygon. We multiply the result found by 200. By the way, such a unit of measurement of angles as degrees is practically not used.
Calculation of external corners of n-gons
For any regular polygon, in addition to the internal one, you can also calculate the external angle. Its value is found in the same way as for other figures. So, to find the outer corner of a regular polygon, you need to know the value of the inner one. Further, we know that the sum of these two angles is always 180 degrees. Therefore, we do the calculations as follows: 180⁰ minus the value of the internal angle. We find the difference. It will be equal to the value of the angle adjacent to it. For example, the inner corner of a square is 90 degrees, so the outer angle will be 180⁰ - 90⁰ = 90⁰. As we can see, it is not difficult to find it. The external angle can take a value from +180⁰ to, respectively, -180⁰.
