These geometric shapes surround us everywhere. Convex polygons can be natural, such as a honeycomb, or artificial (man-made). 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 on the same side of a straight line that passes through a pair of adjacent vertices of this geometric figure. There are other definitions as well. A convex polygon is one that is located in a single half-plane with respect to any straight line containing one of its sides.
Convex polygons
In the course of elementary geometry, only simple polygons are always considered. To understand all the properties of suchgeometric shapes, it is necessary to understand their nature. To begin with, it should be understood that any line is called closed, the ends of which coincide. Moreover, the figure formed by it can have a variety of configurations. A polygon is a simple closed broken line, in which neighboring links are not located on the same straight line. Its links and vertices 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 nth number of vertices, and hence the nth number of sides, is called an n-gon. The broken line itself is called the border or contour of this geometric figure. A polygonal plane or a flat polygon is called the end part of any plane bounded by it. The adjacent sides of this geometric figure are called segments of a broken line emanating from one vertex. They will not be adjacent if they come from different vertices of the polygon.
Other definitions of convex polygons
In elementary geometry, there are several more equivalent definitions indicating which polygon is called convex. All of these statements are equally true. A polygon is considered convex if:
• every segment that connects any two points inside it lies entirely within it;
• inside itall its diagonals lie;
• any internal angle does not exceed 180°.
A polygon always divides 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 is the outer region of this geometric figure. This polygon is an intersection (in other words, a common component) of several half-planes. Moreover, each segment that has ends at 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. So, convex polygons that have an interior angle of 180° are called weakly convex. A convex geometric figure that has three vertices is called a triangle, four - a quadrangle, five - a pentagon, etc. Each of the convex n-gons meets the following essential 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 vertices are located on the same circle, is called inscribed in a circle. A convex polygon is called circumscribed if all its sides near the circle touch it. Two polygons are said to be equal only if they can be superimposed by superposition. A plane polygon is called a polygonal plane.(part of the plane), which is limited by this geometric figure.
Regular convex polygons
Regular polygons are 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. It is called the center of this geometric figure. The segments connecting the center with the vertices of this geometric figure are called apothems, and those that connect point 0 with the sides are called radii.
A regular quadrilateral is a square. An equilateral triangle is called an equilateral triangle. For such figures, there is the following rule: each corner 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=ph, where p is half the sum of all sides of the given polygon and h is the length of the apothem.
Properties of convex polygons
Convex polygons have certain properties. So, a segment that connects any 2 points of such a geometric figure is necessarily located in it. Proof:
Assume 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, which contains any side of P. Therefore, AB also has this property and is contained in P. A convex polygon can always be divided into several triangles by absolutely all the diagonals drawn from one of its vertices.
Angles of convex geometric shapes
The corners of a convex polygon are the corners formed by its sides. Internal corners are located in the inner region of a given 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 angles of a given geometric figure are called external. Each corner of a convex polygon located inside it is:
180° - x, where x is the value of the outer angle. This simple formula works for any geometric shapes of this type.
In general, for external corners there is the following rule: each 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 inside angle is 120°, the outside angle will be 60°.
Sum of angles of convex polygons
The sum of the interior angles of a convex polygon is set 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 quite easy to calculate. Consider any such geometric figure. To determine the sum of the angles inside a convex polygon, it is necessaryconnect one of its vertices to other vertices. As a result of this action, (n-2) triangles are obtained. We know that the sum of the angles of any triangle is always 180°. Since their number in any polygon is (n-2), the sum of the interior 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 equal to 180°. Based on this, you can determine the sum of all its angles:
180 x n.
The sum of the interior angles is 180°(n-2). Based on this, the sum of all external corners of this figure is set by the formula:
180°n-180°-(n-2)=360°.
The sum of the exterior angles of any convex polygon will always be 360° (regardless of the number of sides).
The outer angle of a convex polygon is generally represented by the difference between 180° and the value of the inner 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 also 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, triangles can be very simply made by drawing alldiagonals from one vertex. Thus, any polygon can eventually be divided into a certain number of triangles, which turns out to be very useful in solving various problems associated with such geometric shapes.
Perimeter of a convex polygon
Segments of a broken line, called sides of a polygon, are most often denoted 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 circumference
Convex polygons can be inscribed and circumscribed. A circle that touches all sides of this geometric figure is called inscribed in it. Such a polygon is called circumscribed. The center of a circle that is inscribed in a polygon is the intersection point of the bisectors of all angles within a given geometric figure. The area of such a polygon is:
S=pr, 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 circumscribed around it. Moreover, this convex geometric figure is called inscribed. The center of the circle, which is circumscribed about such a polygon, is the intersection point of the so-called perpendicular bisectors of all sides.
Diagonals of convex geometric shapes
The diagonals of a convex polygon are segments thatconnect non-adjacent vertices. Each of them lies inside this geometric figure. The number of diagonals of such an n-gon is set 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 it is possible to divide each convex polygon 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.
Decomposition of a convex polygon
In some cases, to solve geometric problems, it is necessary to split a convex polygon into several triangles with non-intersecting diagonals. This problem can be solved by deriving a specific formula.
Definition of the problem: let's call a proper partition of a convex n-gon into several triangles by diagonals that intersect only at the vertices of this geometric figure.
Solution: Suppose that Р1, Р2, Р3 …, Pn are vertices of this n-gon. The number Xn is the number of its partitions. Let us carefully consider the obtained diagonal of the geometric figure Pi Pn. In any of the regular partitions P1 Pn belongs to a certain triangle P1 Pi Pn, which has 1<i<n. Proceeding from this and assuming that i=2, 3, 4 …, n-1, we get (n-2) groups of these partitions, which include all possible particular cases.
Let i=2 be one group of regular partitions, always containing the diagonal Р2 Pn. The number of partitions that enter it is the same as the number of partitions(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 Р3 Р1 and Р3 Pn. In this case, 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 a regular partition will certainly contain a triangle P1 P4 Pn, to which the quadrangle P1 P2 P3 P4, (n-3)-gon P4 P5 … Pn will adjoin. The number of regular partitions of such a quadrilateral is X4, and the number of partitions of an (n-3)-gon is Xn-3. Based on the foregoing, we can say that the total number of correct partitions contained in this group is 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 correct splits in this group will be the same as the number of splits in the group where i=2 (in other words, equals Xn-1).
Since X1=X2=0, X3=1, X4=2…, then the number of all partitions of a convex polygon is:
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 + X4X4 + X5 + X6=42
X8=X7 + X6 + X5X4 + X4X5 + X6 + X7=132
Number of correct partitions intersecting one diagonal inside
When checking special cases, one can arrive atthe assumption that the number of diagonals of convex n-gons is equal to 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. Moreover, an (n-3)-quadrilateral can be composed of them. Along with this, each quadrilateral will have a diagonal. Since two diagonals can be drawn in this convex geometric figure, this means that additional (n-3) diagonals can be drawn in any (n-3)-quadrilaterals. Based on this, we can conclude that in any regular partition it is possible to draw (n-3)-diagonals that meet 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. Assume that (Xi. Yi), i=1, 2, 3… n is the sequence of coordinates of all neighboring vertices of a polygon that does not have self-intersections. In this case, its area is calculated using the following formula:
S=½ (∑ (Xi + Xi + 1) (Yi + Yi + 1)), where (X1, Y1)=(Xn +1, Y n + 1).
