In the basic geometry course, it is proved that the sum of the angles of a convex n-gon is 180° (n-2). It turns out that this statement is also true for non-convex polygons.

Theorem 3. The sum of the angles of an arbitrary n-gon is 180° (n - 2).

Proof. Let's divide the polygon into triangles by drawing diagonals (Fig. 11). The number of such triangles is n-2, and in each triangle the sum of the angles is 180°. Since the angles of the triangles are the angles of the polygon, the sum of the angles of the polygon is 180° (n - 2).

Let us now consider arbitrary closed broken lines, possibly with self-intersections A1A2…AnA1 (Fig. 12, a). Such self-intersecting broken lines will be called star-shaped polygons (Fig. 12, b-d).

Let us fix the direction of counting the angles counterclockwise. Note that the angles formed by a closed polyline depend on the direction in which it is traversed. If the direction of the polyline bypass is reversed, then the angles of the polygon will be the angles that complement the angles of the original polygon up to 360°.

If M is a polygon formed by a simple closed broken line passing in a clockwise direction (Fig. 13, a), then the sum of the angles of this polygon will be equal to 180 ° (n - 2). If the broken line is passed in the counterclockwise direction (Fig. 13, b), then the sum of the angles will be equal to 180 ° (n + 2).

In this way, general formula the sum of the angles of a polygon formed by a simple closed polyline has the form \u003d 180 ° (n 2), where is the sum of the angles, n is the number of angles of the polygon, "+" or "-" is taken depending on the direction of bypassing the polyline.

Our task is to derive a formula for the sum of the angles of an arbitrary polygon formed by a closed (possibly self-intersecting) polyline. To do this, we introduce the concept of the degree of a polygon.

The degree of a polygon is the number of revolutions made by a point during a complete sequential bypass of its sides. Moreover, the turns made in the counterclockwise direction are considered with the “+” sign, and the turns in the clockwise direction - with the “-” sign.

It is clear that the degree of a polygon formed by a simple closed broken line is +1 or -1, depending on the direction of the traversal. The degree of the broken line in Figure 12, a is equal to two. The degree of star heptagons (Fig. 12, c, d) is equal to two and three, respectively.

The notion of degree is defined similarly for closed curves in the plane. For example, the degree of the curve shown in Figure 14 is two.

To find the degree of a polygon or curve, you can proceed as follows. Suppose that, moving along the curve (Fig. 15, a), we, starting from some place A1, made a full turn, and ended up at the same point A1. Let's remove the corresponding section from the curve and continue moving along the remaining curve (Fig. 15b). If, starting from some place A2, we again made a full turn and got to the same point, then we delete the corresponding section of the curve and continue moving (Fig. 15, c). Counting the number of remote sections with the signs "+" or "-", depending on their direction of bypass, we obtain the desired degree of the curve.

Theorem 4. For an arbitrary polygon, the formula

180° (n+2m),

where is the sum of the angles, n is the number of angles, m is the degree of the polygon.

Proof. Let the polygon M have degree m and is conventionally shown in Figure 16. M1, …, Mk are simple closed broken lines, passing through which the point makes full turns. A1, …, Ak are the corresponding self-intersection points of the polyline, which are not its vertices. Let us denote the number of vertices of the polygon M that are included in the polygons M1, …, Mk by n1, …, nk, respectively. Since, in addition to the vertices of the polygon M, vertices A1, …, Ak are added to these polygons, the number of vertices of the polygons M1, …, Mk will be equal to n1+1, …, nk+1, respectively. Then the sum of their angles will be equal to 180° (n1+12), …, 180° (nk+12). Plus or minus is taken depending on the direction of bypassing broken lines. The sum of the angles of the polygon M0, remaining from the polygon M after the removal of the polygons M1, ..., Mk, is equal to 180° (n-n1- ...-nk+k2). The sums of the angles of the polygons M0, M1, …, Mk give the sum of the angles of the polygon M, and at each vertex A1, …, Ak we additionally obtain 360°. Therefore, we have the equality

180° (n1+12)+…+180° (nk+12)+180° (n-n1-…-nk+k2)=+360°k.

180° (n2…2) = 180° (n+2m),

where m is the degree of the polygon M.

As an example, consider the calculation of the sum of the angles of a five-pointed asterisk (Fig. 17, a). The degree of the corresponding closed polyline is -2. Therefore, the desired sum of the angles is 180.

A geometric figure composed of segments AB,BC,CD, .., EF, FA in such a way that adjacent segments do not lie on one straight line, and non-adjacent segments do not have common points, is called a polygon. The ends of these segments points A,B,C, D, …, E,F are called peaks polygon, and the segments themselves AB, BC, CD, .., EF, FA - parties polygon.

A polygon is said to be convex if it is on one side of every line that passes through two of its adjacent vertices. The figure below shows a convex polygon:

And the following figure illustrates a non-convex polygon:

The angle of a convex polygon at a given vertex is the angle formed by the sides of this polygon converging at a given vertex. The exterior angle of a convex polygon at some vertex is the angle adjacent to the interior angle of the polygon at the given vertex.

Theorem: The sum of the angles of a convex n-gon is 180˚ *(n-2)

Proof: consider a convex n-gon. To find the sum of all interior angles, we connect one of the vertices of the polygon to other vertices.

As a result, we get (n-2) triangles. We know that the sum of the angles of a triangle is 180 degrees. And since their number in the polygon is (n-2), the sum of the angles of the polygon is 180˚ *(n-2). This is what needed to be proven.

A task:

Find the sum of the angles of a convex a) pentagon b) hexagon c) decagon.

Let's use the formula to calculate the sum of the angles of a convex n-gon.

a) S5 = 180˚*(5-2) = 180˚ *3 = 540˚.

b) S6 180˚*(6-2) = 180˚*4=720˚.

c) S10 = 180˚*(10-2) = 180˚*8 = 1440˚.

Answer: a) 540˚. b) 720˚. c) 1440˚.

Inner corner of a polygon is the angle formed by two adjacent sides of a polygon. For example, ∠ ABC is an interior angle.

External corner of a polygon is the angle formed by one side of the polygon and the extension of the other side. For example, ∠ LBC is the outside corner.

The number of corners of a polygon is always equal to the number of its sides. This applies to both internal and external corners. Although it is possible to construct two equal external corners for each vertex of the polygon, only one of them is always taken into account. Therefore, to find the number of corners of any polygon, one must count the number of its sides.

sum of interior angles

The sum of the interior angles of a convex polygon is equal to the product of 180° and the number of sides without two.

s = 2d(n - 2)

where s is the sum of the angles, 2 d- two right angles (i.e. 2 90 = 180°), and n- the number of sides.

If we swipe from the top A polygon ABCDEF all possible diagonals, then we divide it into triangles, the number of which will be two less than the sides of the polygon:

Therefore, the sum of the angles of the polygon will be equal to the sum of the angles of all the resulting triangles. Since the sum of the angles of each triangle is 180° (2 d), then the sum of the angles of all triangles will be equal to the product of 2 d for their number:

s = 2d(n- 2) = 180 4 = 720°

From this formula it follows that the sum of the interior angles is constant value and depends on the number of sides of the polygon.

Sum of external angles

The sum of the exterior angles of a convex polygon is 360° (or 4 d).

s = 4d

where s is the sum of the outside angles, 4 d- four right angles (i.e. 4 90 = 360°).

The sum of the external and internal angles at each vertex of the polygon is 180° (2 d), since they are adjacent angles. For example, ∠ 1 and ∠ 2 :

Therefore, if the polygon has n parties (and n vertices), then the sum of external and internal angles for all n vertices will be equal to 2 dn. So that from this sum 2 dn to get only the sum of the external angles, it is necessary to subtract the sum of the internal angles from it, that is, 2 d(n - 2):

s = 2dn - 2d(n - 2) = 2dn - 2dn + 4d = 4d

Proof

For the case of a convex n-gon

Let A 1 A 2 . . . A n (\displaystyle A_(1)A_(2)...A_(n)) is a given convex polygon and n> 3 . Then draw from one vertex to opposite vertices ( n− 3) diagonals: A 1 A 3 , A 1 A 4 , A 1 A 5 . . . A 1 A n − 1 (\displaystyle A_(1)A_(3),A_(1)A_(4),A_(1)A_(5)...A_(1)A_(n-1)). Since the polygon is convex, these diagonals divide it into ( n− 2) triangles: Δ A 1 A 2 A 3 , Δ A 1 A 3 A 4 , . . . , Δ A 1 A n − 1 A n (\displaystyle \Delta A_(1)A_(2)A_(3),\Delta A_(1)A_(3)A_(4),...,\Delta A_ (1)A_(n-1)A_(n)). The sum of the angles of the polygon is the same as the sum of the angles of all these triangles. The sum of the angles in each triangle is 180°, and the number of these triangles is n− 2 . Therefore, the sum of the angles n-gon is 180°( n − 2) . The theorem has been proven.

Comment

For a non-convex n-gon, the sum of the angles is also 180°( n− 2) . The proof can be similar, using in addition the lemma that any polygon can be cut by diagonals into triangles, and not relying on the fact that the diagonals are necessarily drawn from one vertex (cutting a non-convex polygon restricted by such a condition is not always possible in the sense that a non-convex polygon does not necessarily have at least one vertex, all of whose diagonals lie inside the polygon, as well as the triangles they form).

These geometric shapes surround us everywhere. Convex polygons are natural, such as honeycombs, or artificial (man-made). These figures are used in the production various kinds coatings, in painting, architecture, decoration, etc. Convex polygons have the property that all their points are on the same side of a line that passes through a pair of adjacent vertices of this line. geometric figure. There are other definitions as well. A polygon is called convex if it is located in a single half-plane with respect to any straight line containing one of its sides.

In the course of elementary geometry, only simple polygons are always considered. To understand all the properties of such, 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 nth number vertices, and hence nth quantity 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 convex polygon is one that has:

Every line segment that connects any two points within it lies entirely within it;

All its diagonals lie inside it;

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 - outer area 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 flat polygon is a polygonal plane (part of a plane), which is limited by this geometric figure.

Regular convex polygons

Regular polygons are geometric shapes with equal angles and parties. 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 the point 0 with the sides are called radii.

A regular quadrilateral is a square. right triangle called equilateral. For such figures, there is the following rule: 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:

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.

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:

Suppose 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 the same side of the line, which 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 break it into several triangles by absolutely all the diagonals that are drawn from one of its vertices.

Angles of convex geometric shapes

The corners of a convex polygon are the corners that are 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. with internal angles of a given geometric figure are called external. Each corner of a convex polygon located inside it is equal to:

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

In general, for exterior angles, there is the following rule: each angle of a convex polygon is equal to the difference between 180° and the value of the interior 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 determined by the formula:

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 angles inside a convex polygon, one of its vertices must be connected 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 180°. Based on this, you can determine the sum of all its angles:

The sum of the interior angles is 180° * (n-2). Based on this, the sum of all external angles of a given figure is determined 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 exterior angle of a convex polygon is generally represented by the difference between 180° and the interior 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 all the diagonals from one vertex. Thus, any polygon, ultimately, can 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

The segments of a broken line, called the sides of a 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 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:

where r is the radius of the inscribed circle and p is the semi-perimeter 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 line segments that connect non-adjacent vertices. Each of them lies inside this geometric figure. The number of diagonals of such an n-gon is determined 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:

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 split a convex polygon into several triangles with non-intersecting diagonals. This problem can be solved by deriving a certain formula.

Definition of the problem: let's call a correct 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 resulting 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

Let i = 2 be one group of regular partitions always containing the diagonal Р2 Pn. The number of partitions included in it coincides with the number of partitions of the (n-1)-gon Р2 Р3 Р4… 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. In this case, the number of regular partitions contained in this group will coincide with the number of partitions of the (n-2)-gon Р3 Р4… 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 quadrilateral 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 for which 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 partitions in this group will match the number of partitions 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 equal to:

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

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 inside

When checking special cases, one can come to the 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. Suppose (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 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).