Convex polygons. Definition of a convex polygon. Diagonals of a convex polygon

These geometric figures surround us everywhere. Convex polygons are natural, for example, bee honeycombs or artificial (created by humans). These figures are used in the production of various types of coatings, in painting, architecture, decorations, etc. Convex polygons have the property that all their points are located on one side of the line that passes through a pair of neighboring vertices of this geometric figure. There are other definitions. Convex is that polygon that is located in a single half-plane with respect to any line containing one of its sides.

Convex polygons

In the course of elementary geometry, we always consider only simple polygons. To understand all the properties of such geometric figures it is necessary to understand their nature. To begin with, it should be understood that any line whose ends coincide is called closed. And the figure formed by it can have a variety of configurations. A polygon is a simple closed polygonal line whose adjacent links do not lie on the same line. Its links and peaks are, respectively, the sides and vertices of this geometric figure. A simple polyline must not have self-intersections.

The vertices of a polygon are called adjacent if they represent the ends of one of its sides. A geometric figure that has the n-th number of vertices, and hence the n-th number of sides, is called an n-gon. The broken line itself is called the boundary or contour of this geometric figure. A polygonal plane or a plane polygon is called the finite part of any plane bounded by it. The neighboring sides of this geometric figure are the segments of a broken line starting from one vertex. They will not be neighboring if they come from different vertices of the polygon.

Other definitions of convex polygons

In elementary geometry, there are several more equivalent definitions in terms of its value, indicating which polygon is called convex. And all these formulations are equally true. A convex polygon is considered to be:

• every segment that connects any two points inside it lies completely in it;

• inside it lie all its diagonals;

• Any internal angle does not exceed 180 °.

The polygon always divides the plane into two parts. One of them is limited (it can be enclosed in a circle), and the other is unlimited. The first is called the inner region, and the second is called the outer region of this geometric figure. This polygon is the intersection (in other words - the common component) of several half-planes. In this case, each segment that has ends at the points that belong to the polygon completely belongs to it.

Varieties of convex polygons

The definition of a convex polygon does not indicate that there are many kinds of them. And each of them has certain criteria. Thus, convex polygons that have an internal angle equal to 180 ° are called weakly convex. A convex geometric figure that has three vertices is called a triangle, four is a quadrangle, five is a pentagon, etc. Each of the convex n-gons meets the following most important requirement: n must be or be greater than 3. Each of the triangles is convex. A geometric figure of this type, in which all the vertices are located on one circle, is called inscribed in a circle. A convex polygon is called described if all its sides near the circle touch it. Two polygons are called equal only if they can be combined by overlapping. A polygonal plane is called a plane polygon (part of the plane), which is limited by this geometric figure.

Correct convex polygons

Correct polygons are geometric figures with equal angles and sides. Within them there is a point 0, which is at the same distance from each of its vertices. It is called the center of this geometric figure. The segments connecting the center with the vertices of this geometric figure are called apophemes, and those that connect the point 0 to the sides are radii.

The right quadrilateral is a square. A regular triangle is called equilateral. For such figures, the following rule exists: every angle of a convex polygon is 180 ° * (n-2) / n,

Where n is the number of vertices of this convex geometric figure.

The area of any regular polygon is defined by the formula:

S = p * h,

Where p is equal to half the sum of all sides of a given polygon, and h is equal to the length of the apophema.

Properties of convex polygons

Convex polygons have certain properties. So, the segment that connects any 2 points of such a geometric figure is necessarily located in it. Evidence:

Suppose that P is a given convex polygon. We take 2 arbitrary points, for example, A, B, which belong to P. According to the existing definition of a convex polygon, these points are located on one side of the line that contains any side of P. Hence, AB also has this property and is contained in P. A convex polygon is always It is possible to split into several triangles by absolutely all the diagonals that are drawn from one of its vertices.

The angles of convex geometric figures

The angles of a convex polygon are the angles that are formed by its sides. Internal corners are in the internal area of this geometric figure. The angle that is formed by its sides, which converge at one vertex, is called the angle of the convex polygon. Angles adjacent to the interior angles of a given geometric figure are called external. Each angle of the convex polygon located inside it is equal to:

180 ° - x,

Where x is the value of the external angle. This simple formula applies to any geometric figures of this type.

In general, for external angles, the following rule exists: every angle of a convex polygon is equal to the difference between 180 ° and the value of the internal angle. It can have values ranging from -180 ° to 180 °. Therefore, when the inner angle is 120 °, the outer angle will be 60 °.

The sum of the angles of convex polygons

The sum of the interior angles of a convex polygon is established by the formula:

180 ° * (n-2),

Where n is the number of vertices of the n-gon.

The sum of the angles of a convex polygon is calculated quite simply. Consider any such geometric figure. To determine the sum of the angles inside a convex polygon, one of its vertices must be connected to other vertices. As a result of this action, we obtain (n-2) triangles. It is known that the sum of the angles of any triangle is always 180 °. Since their number in any polygon equals (n-2), the sum of the internal angles of such a figure is 180 ° x (n-2).

The sum of the angles of a convex polygon, namely any two internal and adjacent external angles, for a given convex geometric figure will always be 180 °. Proceeding from this, it is possible to determine the sum of all its angles:

180 х n.

The sum of the internal angles is 180 ° * (n-2). Proceeding from this, the sum of all external angles of the given figure is established by the formula:

180 ° * n-180 ° - (n-2) = 360 °.

The sum of the outer angles of any convex polygon will always be 360 ° (regardless of the number of its sides).

The outer angle of the convex polygon is generally represented by a difference between 180 ° and the value of the internal angle.

Other properties of a convex polygon

In addition to the basic properties of these geometric figures, they have others that arise when manipulating them. Thus, any of the polygons can be divided into several convex n-gons. For this, it is necessary to continue each of its sides and cut this geometric figure along these straight lines. You can split any polygon into several convex parts and in such a way that the vertices of each of the pieces coincide with all its vertices. From this geometric figure it is very easy to make triangles by holding all the diagonals from one vertex. Thus, any polygon, in the final analysis, can be divided into a certain number of triangles, which is very useful in solving various problems associated with such geometric figures.

Perimeter of a convex polygon

The segments of a broken line, called the sides of a polygon, are most often denoted by the following letters: ab, bc, cd, de, ea. These are the sides of the geometric figure with the vertices a, b, c, d, e. The sum of the lengths of all sides of this convex polygon is called its perimeter.

Circle of a polygon

Convex polygons can be inscribed and described. A circle touching all sides of this geometric figure is called inscribed into it. Such a polygon is called described. The center of the circle that is inscribed in the polygon is the intersection point of the bisectors of all angles inside a given geometric figure. The area of such a polygon equals:

S = p * r,

Where r is the radius of the inscribed circle, and p is the semiperimeter of the given polygon.

A circle containing the vertices of a polygon is called described near it. In this case, this convex geometric figure is called inscribed. The center of the circle, which is described near such a polygon, represents the point of intersection of the so-called middle perpendiculars of all sides.

Diagonals of convex geometric figures

Diagonals of a convex polygon are segments that connect not neighboring vertices. Each of them lies inside this geometric figure. The number of diagonals of such an n-gon is established by the formula:

N = n (n-3) / 2.

The number of diagonals of a convex polygon plays an important role in elementary geometry. The number of triangles (K), into which each convex polygon can be broken, is calculated by the following formula:

K = n - 2.

The number of diagonals of a convex polygon always depends on the number of its vertices.

Splitting a convex polygon

In some cases, to solve geometric problems, it is necessary to break up a convex polygon into several triangles with disjoint diagonals. This problem can be solved by deriving a definite formula.

Definition of the problem: we call a certain decomposition of a convex n-gon into several triangles by diagonals intersecting only at the vertices of this geometric figure.

Solution: Suppose that P1, P2, P3 ..., Pn are the vertices of this n-gon. The number Xn is the number of its partitions. We carefully consider the resulting diagonal of the geometric figure Pi Pn. In any of the regular partitions P1 Pn belongs to a certain triangle P1 Pi Pn, for which 1

Let i = 2 be one group of regular partitions, always containing the diagonal P2 Pn. The number of partitions that enter into it coincides with the number of partitions of the (n-1) -gon P2 P3 P4 ... Pn. In other words, it equals Xn-1.

If i = 3, then this other group of partitions will always contain diagonals P3 P1 and P3 Pn. In this case, the number of regular partitions that are contained in this group will coincide with the number of partitions (n-2) -gon P3 P4 ... Pn. In other words, it will be equal to Xn-2.

Let i = 4, then among the triangles the regular partition necessarily contains a triangle P1 P4 Pn to which the quadrilateral P1 P2 P3 P4, (n-3) -gon P4 P5 ... Pn, adjoins. The number of regular partitions of such a quadrilateral equals X4, and the number of partitions of the (n-3) -gon is equal to Xn-3. Based on all the above, we can say that the total number of regular partitions that are contained in this group is equal to Xn-3 X4. Other groups for which i = 4, 5, 6, 7 ... will contain Xn-4 X5, Xn-5 X6, Xn-6 X7 ... of regular partitions.

Let i = n-2, then the number of regular partitions in the given group will coincide with the number of partitions in the group for which i = 2 (in other words, it equals Xn-1).

Since X1 = X2 = 0, X3 = 1, X4 = 2 ..., then the number of all partitions of a convex polygon is equal to:

Xn = Xn-1 + Xn-2 + Xn-3 X4 + Xn-4 X5 + ... + X 5 Xn-4 + X4 Xn-3 + Xn-2 + Xn-1.

Example:

X5 = X4 + X3 + X4 = 5

X6 = X5 + X4 + X4 + X5 = 14

X7 = X6 + X5 + X4 * X4 + X5 + X6 = 42

X8 = X7 + X6 + X5 * X4 + X4 * X5 + X6 + X7 = 132

The number of regular partitions intersecting one diagonal

In the verification of particular cases, one can come to the assumption that the number of diagonals of convex n-gons equals the product of all partitions of this figure by (n-3).

Proof of this assumption: suppose that P1n = Xn * (n-3), then any n-gon can be decomposed into (n-2) -triangles. At the same time, one of them can be combined (n-3) -the quadrangle. Along with this, each quadrilateral will have a diagonal. Since two diagonals can be drawn in this convex geometric figure, this means that it is possible to draw additional diagonals (n-3) in any (n-3) -ray quadrangles. Proceeding from this, it can be concluded that in any regular partition it is possible to carry out (n-3) -diagonals corresponding to the conditions of this problem.

Area of convex polygons

Often, when solving various problems of elementary geometry, it becomes necessary to determine the area of a convex polygon. Suppose that (Xi. Yi), i = 1,2,3 ... n is a sequence of coordinates of all neighboring vertices of a polygon that does not have self-intersections. In this case, its area is calculated by the following formula:

S = ½ (Σ (X _i + X _{i + 1} ) (Y _i + Y _{i + 1} )),

Where (X ₁ , Y ₁ ) = (X _{n +1} , Y _{n + 1} ).