These geometric shapes surround us everywhere. Convex polygons are natural, for example, bee honeycombs or artificial (created by man). These figures are used in the production of various types of coatings, in painting, architecture, decoration, etc. Convex polygons have the property that all their points are located on one side of the line, which passes through a pair of neighboring vertices of this geometric figure. There are other definitions. Convex is the 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, exclusively simple polygons are always considered. To understand all the properties of such
geometric shapes, you need to understand their nature. To begin with, it should be understood that any line whose ends coincide is called closed. Moreover, the figure formed by her can have a variety of configurations. A polygon is a simple closed polygonal line in which adjacent links are not located on one straight line. Its links and vertices are, respectively, the sides and vertices of this geometric figure. A simple broken line should not have self-intersections.
The vertices of a polygon are called adjacent, if they are the ends of one of its sides. A geometric figure that has an nth number of vertices, and hence an nth number of sides, is called an n-gon. The polyline itself is called the border or contour of this geometric figure. A polygonal plane or a flat polygon is the finite part of any plane bounded by it. The adjacent sides of this geometric figure are the segments of the broken line emanating from one vertex. They will not be adjacent if they emanate from different vertices of the polygon.
Other definitions of convex polygons
In elementary geometry, there are several equivalent definitions that indicate which polygon is called convex. Moreover, all these formulations are equally true. A convex polygon is considered to have:
• each segment that connects any two points inside it lies completely in it;
• inside it are all its diagonals;
• any internal angle does not exceed 180 °.
A polygon always splits a plane into 2 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 - the outer region of this geometric figure. This polygon is the intersection (in other words, the common component) of several half-planes. Moreover, 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. Moreover, each of them has certain criteria. So, convex polygons that have an internal angle of 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 equal to or 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 using overlays. A flat polygon is called a polygonal plane (part of a plane), which is limited by this geometric figure.
Regular Convex Polygons
Regular polygons are called geometric shapes with equal angles and sides. Inside them there is a point 0, which is at the same distance from each of its vertices. She is called the center of this geometric figure. The segments connecting the center with the vertices of this geometric figure are called apofems, and those that connect point 0 with the sides are radii.
The regular quadrangle is a square. A regular triangle is called equilateral. For such figures, the following rule exists: each 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 determined by the formula:
S = p * h,
where p is equal to half the sum of all sides of the given polygon, and h is equal to the length of the apothem.
Convex Polygon Properties
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. Therefore, AB also has this property and is contained in P. A convex polygon is always it is possible to divide into several triangles by absolutely all diagonals that are drawn from one of its vertices.
The angles of convex geometric shapes
The angles of a convex polygon are the angles that are formed by its sides. The inner corners are in the inner region of this geometric figure. The angle that is formed by its sides that converge at one vertex is called the angle of a convex polygon. Angles adjacent to the internal corners of a given geometric shape are called external. Each corner of a 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 shapes of this type.
In general, the following rule exists for external angles: each angle of a convex polygon is equal to the difference between 180 ° and the value of the internal angle. It can range from -180 ° to 180 °. Therefore, when the internal angle is 120 °, the external will be 60 °.
Sum of angles of convex polygons
The sum of the internal angles of a convex polygon is determined 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, you need to connect one of its vertices to other vertices. As a result of this action, an (n-2) triangle is obtained. It is known that the sum of the angles of any triangles is always 180 °. Since their number in any polygon is (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 °. Based on this, you can determine the sum of all its angles:
180 x n.
The sum of the internal angles is 180 ° * (n-2). Based on this, the sum of all the external corners of this figure is determined by the formula:
180 ° * n-180 ° - (n-2) = 360 °.
The sum of the external angles of any convex polygon will always be 360 ° (regardless of the number of sides).
The external angle of a convex polygon in the general case is the 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 shapes, they have others that arise when manipulating them. So, any of the polygons can be divided into several convex n-gons. To do this, it is necessary to continue each of its sides and cut this geometric figure along these straight lines. It is possible to split any polygon into several convex parts in such a way that the vertices of each of the pieces coincide with all its vertices. From such a geometric figure, it is very simple to make triangles by drawing all the diagonals from one vertex. Thus, any polygon, ultimately, can be divided into a certain number of triangles, which is very useful in solving various problems associated with such geometric shapes.
The perimeter of a convex polygon
The segments of the polyline, called the sides of the polygon, are most often indicated by the following letters: ab, bc, cd, de, ea. These are the sides of a geometric figure with vertices a, b, c, d, e. The sum of the lengths of all sides of this convex polygon is called its perimeter.
Polygon circle
Convex polygons can be inscribed and described. The circle touching all sides of this geometric figure is called inscribed in it. Such a polygon is called described. The center of the circle, which is inscribed in the polygon, is the intersection point of the bisectors of all angles inside the given geometric figure. The area of such a polygon is:
S = p * r,
where r is the radius of the inscribed circle, and p is the semiperimeter of the given polygon.
The circle containing the vertices of the polygon is called the circumscribed around it. Moreover, this convex geometric figure is called inscribed. The center of the circle, which is described near such a polygon, is the intersection point of the so-called mid-perpendiculars of all sides.
Diagonals of convex geometric shapes
The diagonals of a convex polygon are the segments that connect non-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 divided 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 divide a convex polygon into several triangles with disjoint diagonals. This problem can be solved by deriving a certain formula.
Definition of the problem: we say that a certain partition of a convex n-gon into several triangles by diagonals intersecting only at the vertices of this geometric figure is correct.
Solution: Suppose that P1, P2, P3 ..., Pn are the vertices of this n-gon. The number Xn is the number of its partitions. 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 <i <n. Based on this and assuming that i = 2,3,4 ..., n-1, we obtain (n-2) groups of these partitions, which include all possible special cases.
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 the diagonals P3 P1 and P3 Pn. Moreover, the number of regular partitions that are contained in this group will coincide with the number of partitions of the (n-2) -gon P3 P4 ... Pn. In other words, it will equal Xn-2.
Let i = 4, then among the triangles the correct partition will certainly contain the triangle P1 P4 Pn, which will adjoin the quadrangle P1 P2 P3 P4, the (n-3) -gon P4 P5 ... Pn. The number of regular partitions of such a quadrilateral is equal to X4, and the number of partitions of a (n-3) -gon is equal to Xn-3. Based on the foregoing, we can say that the total number of regular partitions that are contained in this group equals Xn-3 X4. Other groups with i = 4, 5, 6, 7 ... will contain Xn-4 X5, Xn-5 X6, Xn-6 X7 ... regular partitions.
Let i = n-2, then the number of regular partitions in this group will coincide with the number of partitions in the group for which i = 2 (in other words, equals Xn-1).
Since X1 = X2 = 0, X3 = 1, X4 = 2 ..., 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 that intersect inside one diagonal
When checking particular cases, we can assume 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: imagine that P1n = Xn * (n-3), then any n-gon can be divided into (n-2) -triangles. In this case, an (n-3) -quadagon can be composed of them. Along with this, each quadrangle will have a diagonal. Since two diagonals can be drawn in this convex geometric figure, this means that in any (n-3) -tagon, it is possible to draw additional (n-3) diagonals. Based on this, we can conclude that in any correct partition, it is possible to draw (n-3) -diagonals that meet the conditions of this problem.
The area of convex polygons
When solving various problems of elementary geometry, it is often 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 adjacent vertices of a polygon that does not have self-intersections. In this case, its area is calculated by the following formula:
S = ½ (∑ (X i + X i + 1 ) (Y i + Y i + 1 )),
where (X 1 , Y 1 ) = (X n +1 , Y n + 1 ).