Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, address email etc.
How we use your personal information:
- The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial procedure, legal proceedings, and/or based on public requests or requests from government bodies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
What is a polygon called? Types of polygons. POLYGON, a 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 broken line, consisting of three or more segments (links). A triangle is definitely a polygon. A polygon is a figure that has five or more angles.
Definition. A quadrilateral is a flat geometric figure consisting of four points (the vertices of the quadrilateral) and four consecutive segments connecting them (the sides of the quadrilateral).
A rectangle is a quadrilateral with all right angles. They are named according to the number of sides or vertices: TRIANGLE (three-sided); QUADAGON (four-sided); PENTAGON (five-sided), etc. In elementary geometry, a figure is called a figure bounded by straight lines called sides. The points at which the sides intersect are called vertices. A polygon has more than three angles. This is accepted or agreed upon.
A triangle is a triangle. And a quadrilateral is also not a polygon, and is not called a quadrilateral - it is either a square, a rhombus, or a trapezoid. The fact that a polygon with three sides and three angles has proper name"triangle" does not deprive it of its status as a polygon.
See what “POLYGON” is in other dictionaries:
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. Who hasn't 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.
Although, of course, a figure consisting of three angles 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 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. Note that, as many angles as there are, there are as many sides, so these figures could well be called polylaterals.
Let A1A2...A n be a given convex polygon and n>3. Let's draw diagonals in it (from one vertex)
The sum of the angles of each triangle is 1800, and the number of these triangles n is 2. Therefore, the sum of the angles of the convex n triangle A1A2...A n is 1800* (n - 2). The theorem is proven. External corner convex polygon at a given vertex the angle adjacent to the interior angle of the polygon at this vertex is called.
In a quadrilateral, draw a straight line so that it divides it into three triangles
A quadrilateral never has three vertices on the same line. The word “polygon” indicates that all figures in this family have “many angles.” A broken line is called simple if it has no 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 differently - a regular quadrilateral. Such figures have long been of interest to craftsmen who decorated buildings.
The number of vertices is equal to the number of sides. A polyline is called closed if its ends coincide. They made beautiful patterns, for example on parquet. Our five-pointed star is a regular pentagonal star.
But not all regular polygons could be used to make parquet. Let's take a closer look at two types of polygons: triangle and 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 this lesson we will begin to 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 geometric shapes and have already considered 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 a 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 what is being considered general case the presence of some unknown number of angles (n pieces).
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 the 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 marked external corners
Because The outer corner is connected to the inner one as adjacent, then 
and similarly for the remaining external corners. Then:
During the transformations, we used the already proven theorem about the sum of internal angles of an n-gon.
Proven.
From the proven theorem it follows interesting fact, that the sum of the external angles of a convex n-gon is equal to 
on the number of its angles (sides). By the way, in contrast to the sum of internal angles.
References
- Alexandrov A.D. and others. Geometry, 8th grade. - M.: Education, 2006.
- Butuzov V.F., Kadomtsev S.B., Prasolov V.V. Geometry, 8th grade. - M.: Education, 2011.
- Merzlyak A.G., Polonsky V.B., Yakir S.M. Geometry, 8th grade. - M.: VENTANA-GRAF, 2009.
- Profmeter.com.ua ().
- Narod.ru ().
- Xvatit.com ().
Homework
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 (numbered in the figure 1).
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).
The part of the plane bounded by a closed broken line is called a polygon.
The segments of this broken line are called parties polygon. AB, BC, CD, DE, EA (Fig. 1) are the sides of the polygon ABCDE. The sum of all the sides of a polygon is called its perimeter.
The polygon is called convex, if it is located on one side of any of its sides, indefinitely extended beyond both vertices.
The MNPKO polygon (Fig. 1) will not be convex, since it is located on more than one side of the straight line KR.
We will only consider convex polygons.
The angles formed by two adjacent sides of a polygon are called its internal corners, and their tops are 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 - diagonals of the polygon (Fig. 2).
Angles adjacent to internal corners polygon are called the external angles of the polygon (Fig. 3).
Depending on the number of angles (sides), the polygon is called a triangle, quadrilateral, pentagon, etc.
Two polygons are said to be congruent if they can be brought together by overlapping.
Inscribed and circumscribed polygons
If all the vertices of a polygon lie on a circle, then the polygon is called inscribed into a circle, and the circle - described near the polygon (fig).
If all sides of a polygon are tangent to a 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 number of sides (angles) are called polygons of the same name.
The sides of similar polygons connecting the vertices respectively are called similar. equal angles(rice).
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' ∠C = ∠C' ∠D = ∠D' ∠ E = ∠E' and, in addition, AB / A'B' = BC / B'C' = CD / C'D' = DE / D'E' = EA / E'A' .
Ratio of perimeters of similar polygons
First, consider the property of a series of equal ratios. Let us, for example, have the following ratios: 2 / 1 = 4 / 2 = 6 / 3 = 8 / 4 =2.
Let's find the sum of the previous terms of these relations, then the sum of their subsequent terms and find the ratio of the resulting sums, we get:
$$ \frac(2 + 4 + 6 + 8)(1 + 2 + 3 + 4) = \frac(20)(10) = 2 $$
We get the same thing if we take a series of some other relations, for example: 2 / 3 = 4 / 6 = 6 / 9 = 8 / 12 = 10 / 15 = 2 / 3 Let’s find the sum of the previous terms of these relations and the sum of the subsequent ones, and then 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 both cases, the sum of the previous members of a series of equal relations relates to the sum of subsequent members of the same series, just as the previous member of any of these relations relates to its subsequent one.
We derived this property by considering the series numerical examples. It can be derived strictly and in a general form.
Now consider the ratio of the perimeters of similar 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 derived for a series of equal ratios, we can write:
The sum of the previous terms of the relations we have taken represents the perimeter of the first polygon (P), and the sum of the subsequent terms of these relations represents the perimeter of the second polygon (P’), which means P / P’ = AB / A’B’.
Hence, The perimeters of similar polygons are related to their similar sides.
Ratio of areas of similar polygons
Let ABCDE and A’B’C’D’E’ be similar polygons (Fig).
It is known that ΔАВС ~ ΔA'В'С' 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 series of equal ratios we get:
Or 
where S and S’ are the areas of these similar polygons.
Hence, The areas of similar polygons are related as the squares of similar sides.
The resulting formula can be converted to this form: S / S’ = (AB / A’B’) 2
Area of an arbitrary polygon
Let it be necessary to calculate the area of an arbitrary quadrilateral ABC (Fig.).
Let's draw a diagonal in it, for example AD. We get two triangles ABD and ACD, the areas of which we can calculate. Then we find the sum of the areas of these triangles. The resulting sum will express the area of the given quadrilateral.
If you need to calculate the area of a pentagon, then we do the same thing: we draw diagonals from one of the vertices. We get three triangles, the areas of which we can calculate. This means we can find the area of this pentagon. We do the same when calculating the area of any polygon.
Projected area of a polygon
Let us recall that the angle between a line and a plane is the angle between a given line and its projection onto the plane (Fig.).
Theorem. The area of the orthogonal projection of a polygon onto a 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 projection plane.
Each polygon can be divided into triangles whose sum of areas is equal to the area of the polygon. Therefore, it is enough to prove the theorem for a triangle.
Let ΔАВС be projected onto the plane r. Let's consider two cases:
a) one of the sides ΔABC is parallel to the plane r;
b) none of the sides ΔABC are parallel r.
Let's consider first case: let [AB] || r.
Let us draw a plane through (AB) r 1 || r and project orthogonally ΔАВС on r 1 and on r(rice.); we get ΔАВС 1 and ΔА'В'С'.
By the property of projection we have ΔАВС 1 (cong) ΔА'В'С', and therefore
S Δ ABC1 = S Δ A’B’C’
Let's draw ⊥ and the segment D 1 C 1 . Then ⊥ , a \(\overbrace(CD_1C_1)\) = φ is the value of the angle between the plane ΔABC 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 consider second case. Let's draw a plane r 1 || r through that vertex ΔАВС, the distance from which to the plane r the smallest (let this be vertex A).
Let's project ΔАВС on the plane r 1 and r(rice.); let its projections be ΔАВ 1 С 1 and ΔА'В'С', respectively.
Let (BC) ∩ 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 materials
