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.
Regular polygon properties
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 shape. In addition, a circle can be inscribed into 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 geometrical figures are subject to the same theorems. Any side of a regular n-gon is related to the radius of the circumscribed circle R. 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 number of equal segments, which, when connected, form closed line... In this case, all the angles 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 parties. They also include star-shaped figures. For complex regular polygons, the sides are found by inscribing them into a circle. Here is a proof. Draw a regular polygon with an arbitrary number of sides n. Draw a circle around it. Set the radius R. Now imagine that you are given some n-gon. If the points of its corners 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 regular triangle
An equilateral triangle is a regular polygon. The formulas apply to it the same as to the square and n-gon. A triangle will be considered correct if it has sides of the same length. In this case, the angles are equal to 60⁰. Let's construct a triangle with a given side length a. Knowing its median and height, you can find the meaning of its sides. To do this, we will use the method of finding through the formula a = 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 will be 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. In this case, it must be projected strictly onto the base of the figure. So, knowing the height x, we find the side a of an isosceles triangle by the formula a = b = 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 of the base c: 2 = √ (x: cosα) ^ 2 - (x ^ 2) = √x ^ 2 (1 - cos ^ 2α): cos ^ 2α = x ∙ tgα. Then c = 2xtgα. 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 = b = c = q = e ∙ cosα = e√2: 2, where e is the diagonal of the square, or the base of the right-angled triangle formed after dividing. This is not the only way to find the sides of a square. Let's inscribe this shape into 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 = 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 not difficult to calculate it. To do this, you need to know the meanings of all parties. There are special formulas for some types of polygons. 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 shape. In general, it looks like this: P = 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, it is necessary 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, one is enough. By the same principle, we find the perimeter of the quadrangles, that is, the square and the 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 give an example. If the side of a rhombus or square is 6 cm, then we find the perimeter as follows: P = 4 ∙ 6 = 24 cm. Only opposite sides of a parallelogram are equal. Therefore, its perimeter is found using a different method. So, we need to know the length a and the width in the figure. Then we apply the formula P = (a + b) ∙ 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-angled triangle
The perimeter of the correct one can be found by the formula P = 3a, where a is the length of the side. If it is unknown, it can be found through the median. In a right-angled triangle, only two sides are of equal importance. The foundation 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 = a + b + c, where a and b are equal sides, and c is the base. Recall that in an isosceles triangle a = b = a, so a + b = 2a, then P = 2a + c. For example, if the side of an isosceles triangle is 4 cm, we will find its base and perimeter. We calculate the value of the hypotenuse according to the Pythagorean theorem c = √a 2 + in 2 = √16 + 16 = √32 = 5.65 cm. Now we calculate the perimeter P = 2 ∙ 4 + 5.65 = 13.65 cm.
How to find the corners of a regular polygon
A regular polygon occurs in our life 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 build any n-gon, you need to know the value of its angles. But how do you find them? Even ancient scientists tried to build regular polygons. They guessed to inscribe them in circles. And then they marked the necessary points on it, connected them with straight lines. For simple shapes, the construction problem has been resolved. Formulas and theorems have been obtained. For example, Euclid in his famous work "Inception" was engaged in solving problems for 3-, 4-, 5-, 6- and 15-gons. He found ways to construct them and find the corners. Let's see how to do this for a 15-gon. First, you need to calculate the sum of its interior angles. You must use the formula S = 180⁰ (n-2). So, we are given a 15-gon, which means that the number n is 15. Substitute the data we know into the formula and we get S = 180⁰ (15 - 2) = 180⁰ x 13 = 2340⁰. We have found the sum of all the interior angles of a 15-gon. Now you need to get the value of each of them. There are 15 angles in total. We do the calculation 2340⁰: 15 = 156⁰. This means that each internal angle is 156⁰, now with the help of a ruler and a compass, you can build a regular 15-gon. But what about more complex n-gons? For many centuries, scientists have struggled to solve this problem. It was found only in the 18th century by Karl Friedrich Gauss. He was able to build a 65537-gon. Since then, the problem is officially considered completely resolved.
Calculating the 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? You must proceed as follows. First, we find out the number of sides of a regular polygon, then subtract 2. So we get the value: n - 2. Multiply the found difference by the number n ("pi" = 3.14). Now all that remains is to divide the resulting product by the number of angles in the n-gon. Consider these calculations using the example of the same hexagon. So, the number n is 15. Let's apply the formula S = n (n - 2): n = 3.14 (15 - 2): 15 = 3.14 ∙ 13: 15 = 2.72. This is, of course, not the only way to calculate the angle in radians. You can simply divide the angle in degrees by 57.3. After all, exactly this many degrees is equivalent to one radian.
Calculating 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 degrees. This is done as follows. Subtract 2 from the total number of angles, divide the resulting difference by the number of sides of a regular polygon. We multiply the found result by 200. By the way, such a unit of measurement of angles as degrees is practically not used.
Calculation of external angles of n-gons
For any regular polygon, besides the inner one, you can also calculate the outer angle. Its meaning is found in the same way as for the rest of the 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. Find the difference. It will be equal to the value of the adjacent angle. For example, the inside corner of the square is 90 degrees, so the outside 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 -180⁰, respectively.
Types of polygons:
Quadrangles
Quadrangles, respectively, consist of 4 sides and corners.
Sides and corners opposite each other are called opposite.
Diagonals divide convex quadrangles into triangles (see picture).
The sum of the angles of a convex quadrilateral is 360 ° (according to the formula: (4-2) * 180 °).
Parallelograms
Parallelogram is a convex quadrangle with opposite parallel sides (in the figure under number 1).
Opposite sides and angles in a parallelogram are always equal.
And the diagonals at the intersection are halved.
Trapeze
Trapezoid is also a quadrangle, and in trapezium only two sides are parallel, which are called grounds... Other parties are lateral 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 corners is straight, then the trapezoid is rectangular.
The middle line of the trapezoid is equal to the half-sum of the bases and is 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 - diagonals of the rhombus are perpendicular to each other and bisect the corners of the rhombus.
In the figure, rhombus number 5.
Rectangles
Rectangle is a parallelogram, each corner of which is a straight line (see figure 8).
In addition to the properties of a parallelogram, rectangles have their own special property - the diagonals of the rectangle are.
Squares
Square is a rectangle with all sides equal (# 4).
Has the properties of a rectangle and a rhombus (since all sides are equal).
What is called a polygon? Types of polygons. POLYGON, flat 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 segments (sides of a quadrilateral) connecting them.
A rectangle is a rectangle with all 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 where the sides meet 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 quadrangle, 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 a 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.
Subject, 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 a study of the number of constituent elements of regular polygons (from a triangle to n - a 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; to develop research and cognitive activities;
Educational task: to bring up independence, activity, responsibility for the assigned work, perseverance in achieving the set goal.
During the classes: a quote is written 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 a plane?
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: obtaining 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 along the sides (versatile, isosceles, equilateral) and corners (acute-angled, obtuse, 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 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 smallest number sides of the polygon - 3. But the triangles, connecting with each other, can form other shapes, which in turn are also polygons.
The lines connecting non-adjacent vertices of the 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 a 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 135 0. And if any point is the vertex of two such octagons, then their share will be 270 0, and the third octagon has nowhere 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?
Group 2
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.
Students work with the text, look 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, the 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 | Degree measure outside angle | 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 the interior corners of 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.
§ 1 The concept of a triangle
In this lesson, you will become familiar with shapes such as triangle and polygon.
If three points that do not lie on one straight line are connected by segments, then you get a triangle. The triangle has three vertices and three sides.
Before you triangle ABC, it has three vertices (point A, point B and point C) and three sides (AB, AC and CB).
By the way, these same sides can be called in another way:
AB = BA, AC = CA, CB = BC.
The sides of the triangle form three angles at the vertices of the triangle. In the picture you can see angle A, angle B, angle C.
Thus, a triangle is a geometric figure formed by three segments that connect three points that do not lie on one straight line.
§ 2 The concept of a polygon and its types
Besides triangles, there are quadrangles, pentagons, hexagons, and so on. In one word, they can be called polygons.
In the figure, you can see the DMKE quadrilateral.
Points D, M, K and E are the vertices of the quadrilateral.
The segments DM, MK, KE, ED are the sides of this quadrangle. Just as in the case of a triangle, the sides of a quadrilateral form four corners at the vertices, as you guessed it, hence the name - quadrilateral. For this quadrilateral, you can see in the picture angle D, angle M, angle K and angle E.
What quadrangles do you already know?
Square and rectangle! Each of them has four corners and four sides.
Another type of polygons is the pentagon.
Points O, P, X, Y, T are the vertices of the pentagon, and the segments TO, OP, PX, XY, YT are the sides of this pentagon. The pentagon has five corners and five sides, respectively.
How many angles and sides do you think a hexagon has? That's right, six! Reasoning in a similar way, you can tell how many sides, vertices, or corners a particular polygon has. And we can conclude that a triangle is also a polygon, which has exactly three corners, three sides and three vertices.
Thus, in this lesson, you got acquainted with such concepts as a triangle and a polygon. We learned that a triangle has 3 vertices, 3 sides and 3 corners, a quadrangle - 4 vertices, 4 sides and 4 corners, a pentagon - respectively 5 sides, 5 vertices, 5 corners and so on.
List of used literature:
- Mathematics grade 5. Vilenkin N.Ya., Zhokhov V.I. et al. 31st ed., erased. - M: 2013.
- Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
- We calculate without errors. Works with self-test in mathematics 5-6 grades. Author - Minaeva S.S. - year 2014
- Didactic materials in mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
- Control and independent work in mathematics grade 5. Authors - Popov M.A. - year 2012
- Mathematics. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Erased. - M .: Mnemosina, 2009
