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A polygon is called geometric figure, which is bounded on all sides by a closed broken line. In this case, the number of links of the broken line should not be less than three. Each pair of broken line segments has common point and forms angles. The number of angles together with the number of polyline segments are the main characteristics of a polygon. In each polygon, the number of links of the bounding closed polygon coincides with the number of angles.
In geometry, sides are usually called the links of a broken line that limits a geometric object. Vertices are the points of contact between two adjacent sides., from the number of which polygons get their names.
If a closed broken line consists of three segments, it is called a triangle; accordingly, from four segments - a quadrangle, from five - a pentagon, etc.
To designate a triangle or quadrilateral, capital Latin letters are used to designate its vertices. The letters are named in order - clockwise or counterclockwise.
Basic Concepts
When describing the definition of a polygon, you should consider some related geometric concepts:
- If the vertices are the ends of one side, they are called adjacent.
- If a segment connects non-adjacent vertices, then it is called a diagonal. A triangle cannot have diagonals.
- An internal angle is the angle at one of the vertices, which is formed by its two sides converging at this point. It is always located in the inner region of the geometric figure. If the polygon is non-convex, its size may exceed 180 degrees.
- An external angle at a certain vertex is an angle adjacent to the internal one at the same vertex. In other words, the external angle can be considered the difference between 180° and the value of the internal angle.
- The sum of the values of all segments is called the perimeter.
- If all sides and all angles are equal, it is called correct. Only convex ones can be correct.
As mentioned above, the names of polygonal geometric ones are based on the number of vertices. If a figure has n numbers of them, it is called n-gon:
- A polygon is called planar if it limits the finite part of the plane. This geometric figure can be inscribed in a circle or circumscribed around a circle.
- An n-gon is called convex if it meets one of the conditions given below.
- The figure is located on one side of a straight line that connects two adjacent vertices.
- This figure serves as a common part or intersection of several half-planes.
- The diagonals are located inside the polygon.
- If the ends of a segment are located at points that belong to a polygon, the entire segment belongs to it.
- A figure can be called regular if all its segments and all angles are equal. Examples are a square, an equilateral triangle, or a regular pentagon.
- If an n-gon is non-convex, all its sides and angles are equal, and its vertices coincide with those of a regular n-gon, it is called stellated. Such figures may have self-intersections. Examples would be a pentagram or a hexagram.
- A triangle or quadrilateral is said to be inscribed in a circle when all its vertices are located inside one circle. If the sides of this figure have points of contact with the circle, it is a polygon circumscribed about a certain circle.
Any a convex n-gon can be divided into triangles. In this case, the number of triangles is less than the number of sides by 2.
Types of figures
It is a polygon with three vertices and three line segments connecting them. In this case, the connecting points of the segments do not lie on the same straight line.
The connection points of the segments are vertices of the triangle. The segments themselves are called sides of the triangle. The total sum of the interior angles of each triangle is 180°.
According to the relationships between the sides, all triangles can be divided into several types:
- Equilateral- in which the length of all segments is the same.
- Isosceles- triangles in which two of the three segments are equal.
- Versatile- if the length of all segments is different.
In addition, it is customary to distinguish the following triangles:
- Acute-angled.
- Rectangular.
- Obtuse.
Quadrangle
A quadrilateral is a flat figure that has 4 vertices and 4 segments that connect them in series.
- If all the angles of a quadrilateral are right angles, this figure is called a rectangle.
- A rectangle whose sides are all the same size is called a square.
- A quadrilateral whose sides are all equal is called a rhombus.
There cannot be three vertices of a quadrilateral on one straight line.
Video
For more information about polygons, watch this video.
Knowledge of terminology, as well as knowledge of the properties of various geometric shapes will help in solving many problems in geometry. When studying a section such as planimetry, a student often comes across the term “polygon”. What figure does this concept characterize?
Polygon - definition of a geometric figure
A closed broken line, all sections of which lie in the same plane and have no sections of self-intersection, forms a geometric figure called a polygon. The number of links of the broken line 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.
When solving problems involving a polygon, concepts such as:
- Side of a polygon. This term characterizes a segment (link) of a broken chain of the desired figure.
- Polygon angle (internal) – an angle that is formed by 2 adjacent links of a broken line.
- The vertex of a polygon is defined as the vertex of a polyline.
- A polygon diagonal is a segment connecting any 2 vertices (except adjacent ones) of a polygonal figure.
In this case, the number of links and the number of vertices of the broken line within one polygon coincide. Depending on the number of angles (or polyline segments, respectively), the type of polygon is determined:
- 3 angles - triangle.
- 4 corners - a quadrilateral.
- 5 corners - pentagon, etc.
If a polygonal figure has equal angles and accordingly the sides, then they say that this 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 figure itself, you have a convex polygonal figure.
- If, after continuing a side (any one), the resulting straight line intersects the polygon, we're talking about about a concave polygon.
Polygon Properties
Regardless of whether the polygonal figure being studied is regular or not, it has the following properties. So:
- Its internal angles form a total of (p – 2)*π, where
π – radian measure of the rotated angle, corresponds to 180°,
p – 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 – number of sides of a p-gon.
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 of geometric figures and have already examined the simplest of them: triangles and circles. At the same time, we also discussed specific special cases of these figures, such as right, isosceles and regular triangles. Now it's time to talk about more general and complex figures - polygons.
With a special case polygons we are already familiar - this is a triangle (see Fig. 1).
Rice. 1. Triangle
The name itself already emphasizes that this is a figure with three angles. Therefore, in polygon there can be many of them, i.e. more than three. For example, let’s draw a pentagon (see Fig. 2), i.e. figure with five corners.
Rice. 2. Pentagon. Convex polygon
Definition.Polygon- a figure consisting of several points (more than two) and the corresponding number of segments that sequentially connect them. These points are called peaks polygon, and the segments are parties. In this case, no two adjacent sides lie on the same straight line and no two non-adjacent sides intersect.
Definition.Regular polygon is a convex polygon in which all sides and angles are equal.
Any polygon divides the plane into two areas: internal and external. The internal area is also referred to as polygon.
In other words, for example, when they talk about a pentagon, they mean both its entire internal region and its border. And the internal region includes all points that lie inside the polygon, i.e. the point also refers to the pentagon (see Fig. 2).
Polygons are also sometimes called n-gons to emphasize that the general case of the presence of some unknown number of angles (n pieces) is considered.
Definition. Polygon perimeter- the sum of the lengths of the sides of the polygon.
Now we need to get acquainted with the types of polygons. They are divided into convex And non-convex. For example, the polygon shown in Fig. 2 is convex, and in Fig. 3 non-convex.
Rice. 3. Non-convex polygon
Definition 1. Polygon called convex, if when drawing a straight line through any of its sides, the entire polygon lies only on one side of this straight line. Non-convex are everyone else polygons.
It is easy to imagine that when extending any side of the pentagon in Fig. 2 it will all be on one side of this straight line, i.e. it is convex. But when drawing a straight line through a quadrilateral in Fig. 3 we already see that it divides it into two parts, i.e. it is not convex.
But there is another definition of the convexity of a polygon.
Definition 2. Polygon called convex, if when choosing any two of its interior points and connecting them with a segment, all points of the segment are also interior points of the polygon.
A demonstration of the use of this definition can be seen in the example of constructing segments in Fig. 2 and 3.
Definition. Diagonal of a polygon is any segment connecting two non-adjacent vertices.
To describe the properties of polygons, there are two most important theorems about their angles: theorem on the sum of interior angles of a convex polygon And theorem on the sum of exterior angles of a convex polygon. Let's look at them.
Theorem. On the sum of interior angles of a convex polygon (n-gon).
Where is the number of its angles (sides).
Proof 1. Let us depict in Fig. 4 convex n-gon.
Rice. 4. Convex n-gon
From the vertex we draw all possible diagonals. They divide an n-gon into triangles, because each of the sides of the polygon forms a triangle, except for the sides adjacent to the vertex. It is easy to see from the figure that the sum of the angles of all these triangles will be exactly equal to the sum of the internal angles of the n-gon. Since the sum of the angles of any triangle is , then the sum of the internal angles of an n-gon is:
Q.E.D.
Proof 2. Another proof of this theorem is possible. Let's draw a similar n-gon in Fig. 5 and connect any of its interior points with all vertices.
Rice. 5.
We have obtained a partition of the n-gon into n triangles (as many sides as there are triangles). The sum of all their angles is equal to the sum of the interior angles of the polygon and the sum of the angles at the interior point, and this is the angle. We have:
Q.E.D.
Proven.
According to the proven theorem, it is clear that the sum of the angles of an n-gon depends on the number of its sides (on n). For example, in a triangle, and the sum of the angles is . In a quadrilateral, and the sum of the angles is, etc.
Theorem. On the sum of external angles of a convex polygon (n-gon).
Where is the number of its angles (sides), and , …, are the external angles.
Proof. Let us depict a convex n-gon in Fig. 6 and designate its internal and external angles.
Rice. 6. Convex n-gon with designated external angles
Because The 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.
Bibliography
- 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
Polygon is a geometric figure bounded by a closed broken line that has no self-intersections.
The links of the broken line are called sides of the polygon, and its vertices - vertices of the polygon.
Angles of a polygon are the interior angles formed by adjacent sides. The number of angles of a polygon is equal to the number of its vertices and sides.
Polygons are named according to the number of sides. The polygon with the fewest sides is called a triangle; it has only three sides. A polygon with four sides is called a quadrilateral, with five sides a pentagon, etc.
The designation of a polygon is made up of the letters standing at its vertices, naming them in order (clockwise or counterclockwise). For example, they say or write: pentagon ABCDE :
In a pentagon ABCDE points A, B, C, D And E are the vertices of the pentagon, and the segments AB, B.C., CD, DE And E.A.- sides of a pentagon.
Convex and concave
The polygon is called convex, if none of its sides, extended to a straight line, intersects it. Otherwise, the polygon is called concave:
Perimeter
The sum of the lengths of all sides of a polygon is called its perimeter.
Polygon perimeter ABCDE is equal to:
AB + B.C.+ CD + DE + E.A.
If a polygon has all sides and all angles equal, then it is called correct. Regular polygons There can only be convex polygons.
Diagonal
Diagonal of a polygon- this is a segment connecting the vertices of two angles that do not have a common side. For example, a segment AD is the diagonal:
The only polygon that does not have a single diagonal is a triangle, since it has no angles that do not have common sides.
If all possible diagonals are drawn from any vertex of a polygon, they will divide the polygon into triangles:
There will be exactly two fewer triangles than sides:
t = n - 2
Where t is the number of triangles, and n- number of sides.
Dividing a polygon into triangles using diagonals is used to find the area of the polygon, since to find the area of a polygon, you need to divide it into triangles, find the area of these triangles and add the results obtained.
