The rhombus is a regular polygon. regular polygon

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Definition regular polygon may depend on the definition of a polygon: if it is defined as a flat closed polyline, then the definition appears regular star polygon as non-convex a polygon in which all sides are equal and all angles are equal.

Properties

Coordinates

Let be x C (\displaystyle x_(C)) and y C (\displaystyle y_(C)) are the coordinates of the center, and R (\displaystyle R)- radius of the circle, ϕ 0 (\displaystyle (\phi )_(0))- the angular coordinate of the first vertex, then the Cartesian coordinates of the vertices of a regular n-gon are determined by the formulas:

where i = 0 … n ​​− 1 (\displaystyle i=0\dots n-1)

Dimensions

Let be R (\displaystyle R)- the radius of the circle described around the regular polygon, then the radius of the inscribed circle is equal to

and the side length of the polygon is

Square

N (\displaystyle n) and side length a (\displaystyle a) is:

Area of ​​a regular polygon with number of sides n (\displaystyle n), inscribed in a circle of radius R (\displaystyle R), is:

Area of ​​a regular polygon with number of sides n (\displaystyle n) circumscribed around a circle of radius r (\displaystyle r), is:

Area of ​​a regular polygon with number of sides n (\displaystyle n) is equal to

where r (\displaystyle r)- distance from the middle of the side to the center, a (\displaystyle a)- side length.

The area of ​​a regular polygon in terms of the perimeter ( P (\displaystyle P)) and the radius of the inscribed circle ( r (\displaystyle r)) is:

Perimeter

If you need to calculate the side length of a regular n-gon inscribed in a circle, knowing the circumference of the circle L (\displaystyle L) you can calculate the length of one side of the polygon:

Perimeter P n (\displaystyle P_(n)) equals

where n (\displaystyle n) is the number of sides of the polygon.

Application

Regular polygons are by definition the faces of regular polyhedra.

Ancient Greek mathematicians (Antiphon, Bryson of Heracles, Archimedes, etc.) used regular polygons to calculate a number. They calculated the areas of polygons inscribed in a circle and circumscribed around it, gradually increasing the number of their sides and thus obtaining an estimate of the area of ​​the circle.

Story

Construction of a regular polygon with n sides remained a problem for mathematicians well into the 19th century. Such a construction is identical to the division of the circle into n equal parts, since by connecting the points dividing the circle into parts, you can get the desired polygon.

Since then, the problem has been considered completely solved.

A polygon is called regular if all its sides and all angles are equal. Among triangles, an equilateral triangle and only it will be correct. A square (and only a square) is a regular quadrilateral. Let us show that there are regular polygons with any number of sides , where . To do this, we present two methods for constructing such polygons.

Method 1. Take an arbitrary circle and divide it into equal parts. Such a construction is by no means feasible with a compass and straightedge, but we will assume here that such a construction has been done. We take the division points in their sequential position on the circle as the vertices of a -gon inscribed in this circle. Let us prove that the constructed -gon is regular. Indeed, the sides of our polygon (Fig. 312) are chords subtracted by equal arcs, and therefore they are equal to each other.

All angles are based on equal arcs and therefore are also equal. So the polygon is correct.

Method 2. Again, divide the circle into equal parts and draw tangents to the circle at the division points; we restrict each of the tangents by the points of its intersection with the tangents drawn at the adjacent division points. We get a regular polygon circumscribed about a circle (Fig. 313). In fact, its angles are all equal, since each of them, like the angle between tangents, is measured by the half-difference of the arcs, of which the smaller is always equal to a part of the circle, and the larger is always equal to the full circle minus the part. The equality of the sides can be seen at least from the equality of triangles formed by pairs of semi-tangents and chords (for example, triangles, etc.). All of them are isosceles, have equal angles at the vertices and equal bases.

Two regular -gons with the same number sides are similar.

Indeed, their sides are certainly in a constant relationship, equal to the ratio of any pair of sides. In addition, by the theorem on the sum of angles of a -gon, every regular -gon has the same angles equal to 1. The conditions of the criterion of item 224 are satisfied, and -gons are similar.

So, for every regular -gons are similar. From this we immediately obtain a number of corollaries:

1. Two regular -gons with equal parties are equal.

2. A circle can be circumscribed around any regular -gon.

Proof. Take any regular polygon with the same number of sides as the given one, constructed according to the first method, i.e., inscribed in a circle. Let's transform it similarly so that it becomes equal to the given one. Then the circle circumscribed around it is similarly transformed into a circle circumscribed around the given polygon.

3. A circle can be inscribed in every regular polygon.

The proof is similar. It is useful, however, to carry out the reasoning somewhat differently. We already know that a circle can be circumscribed around a given polygon. Let's take its center. The sides of a polygon serve as its chords; being equal to each other, they must be equally spaced from the center. Therefore, a circle with the same center and radius, equal to the distance from the center to the sides of the polygon, will touch all sides of the polygon, i.e. it will be an inscribed circle.

So, the inscribed and circumscribed circles of a regular polygon have a common center. It is called the center of the given regular polygon. The radius of the circumscribed circle is called the radius of the polygon, the radius of the inscribed circle is called its apothem. It is clear that the apothem is always less than the radius.

Regular polygons

In the textbook "Geometry 7-11" by A.V. Pogorelov (18), the topic "Regular polygons" is studied in § 13 "Polygons" p. 115.

The definition of a "regular polygon" is considered at the beginning of the paragraph: "A convex polygon is called regular if all its sides are equal and all angles are equal." Then the definitions of "inscribed" and "circumscribed" polygons are given and the theorem is considered: "A regular convex polygon is inscribed in a circle and circumscribed about a circle."

In the textbook "Geometry 7-9" by L.S. Atanasyan (4), the topic "Regular polygons" is considered in paragraph 105 § 1 "Regular polygons" of Chapter 12.

The definition of a "regular polygon" is given at the beginning of the paragraph:

"A regular polygon is a convex polygon in which all angles are equal and all sides are equal." Then a formula is derived for calculating the angle b n of a regular n-gon:

In the textbook "Geometry 7-9" by I.M. Smirnova, V.A. Smirnov, the "regular polygon" is studied in paragraph 6 "Polygons and polygons".

At the beginning of the paragraph, the definition of “broken line” is introduced: “A figure formed by segments located so that the end of the first is the beginning of the second, the end of the second is the beginning of the third, etc., is called a broken line or simply a broken line.”

Then the definitions of a simple, closed, and polygon are given: "A polygonal line is called simple if it has no points of self-intersection." “If the beginning of the first segment of the polyline coincides with the end of the last one, then the polyline is called closed.” "A figure formed by a simple closed broken line and a part of the plane bounded by it is called a polygon."

After that, the definition of a “regular polygon” is considered: “A polygon is called regular if all its sides and all angles are equal.”

Consider the methodology for studying the topic "Regular polygons" using the example of A.V. Pogorelov's geometry textbook.

At the beginning of the paragraph, the definition of a “regular polygon” is introduced: “A convex polygon is called regular if all its sides are equal and all angles are equal”, then the definitions of “inscribed” and “circumscribed” polygons are introduced: “A polygon is called inscribed in a circle if all of its the vertices lie on some circle”; "A polygon is said to be inscribed about a circle if all its sides are tangent to some circle."

Before studying Theorem 13.3, in order to prepare the class for the proof, you can ask students questions for repetition:

Which line is called tangent to the circle?

What is the relationship between the line and the circle? There is a discussion in the class, which consists of two parts: first

we are talking about a circle circumscribed about a polygon, and then about a circle inscribed in a polygon.

Students' answers are accompanied by a sequential display of a series of drawings.

Which triangle is called inscribed in a circle or which circle is called circumscribed near a triangle (Fig. 1)?

Is it possible to circumscribe a circle around an arbitrary triangle?

How to find the center of a circle circumscribed about a triangle? (Fig.2) What is the radius? (Fig.3)

Is it always possible to describe a circle around a polygon? (No. Example: rhombus if it is not a square. Fig.4)

Is it possible to describe a circle around a regular polygon? (Fig.5)

The first part of Theorem 13.3 is formulated. It is assumed that a circle can be circumscribed around a regular polygon. It is worth noting that this fact will be proved later.

Similar work is being done on the possibility of inscribing a circle in a polygon. The class has the same 5 questions about a circle inscribed in a polygon. At the same time, by analogy with the first part of the conversation, a series of drawings similar to the previous ones is used.

The teacher draws the attention of students to the possibility of inscribing a circle in a regular polygon. Theorem 13.3 is formulated and proved: "A regular convex polygon is inscribed in a circle and circumscribed about a circle."

The proof of the theorem is carried out according to the textbook. It is useful to emphasize that the centers of the inscribed and circumscribed circles in a regular polygon coincide and given point called the center of the polygon.

After proving the theorem, the following tasks are proposed:

1. The side of a regular triangle inscribed in a circle is equal to a. Find the side of the square inscribed in this circle.

Given: Circle (0;R),

DAVS - correct, inscribed,

CMRE - inscribed square.

DAVS - correct, inscribed: R = KMPE - inscribed square in a circle (0;R).

Let x \u003d KM - side of the square, then

Answer: KM = .

2. A regular triangle is inscribed in a circle with a radius of 4 dm, on the side of which a square is built. Find the radius of the circle circumscribing the square.

Given: circle (0;R),

DAVS - correct, inscribed,

Okr. 1 (O;R 1),

ABDE - inscribed square in Okr. one

Find: R 1 .

1. DAVS - correct, inscribed:

ABDE - inscribed square in Okr. one:

Answer: dm.

3. The side of a regular polygon is a, and the radius of the circumscribed circle is R. Find the radius of the inscribed circle. Given: Env.(0;R),

A 1 A 2 ...A n - correct, inscribed,

A 1 A 2 =a , radius=R,

OS is the radius of the inscribed circle.

OS 2 = OB 2 - BC 2

Answer: OS=.

4. The side of a regular polygon is equal to a, and the radius of the inscribed circle is r. Find the radius of the circumscribed circle.

Given: circumference(0;r),

A 1 A 2 ...A n - correct, described,

A 1 A 2 \u003d a, radius \u003d r,

Circle (0;R).

Decision. OB is the radius of the circumscribed circle.

DOSV - rectangular (ZC = 90°)

OB 2 \u003d OS 2 + SW 2

Answer: R = .

Then students can be given a system of tasks:

1. In a regular hexagon A 1 A 2 A 3 A 4 A 5 A 6, the side is 8. The segment BC connects the midpoints of the sides A 3 A 4 and A 5 A b. Find the length of the segment connecting the midpoint of side A 1 A 2 with the midpoint of segment BC.

2. The side of the regular hexagon ABCDEF is equal to 32. Find the radius of the circle inscribed in the triangle MRK if M, P and K are the midpoints of the sides AB, CD. EF respectively.

Express side b of a regular circumscribed polygon in terms of radius R of the circle and side a of a regular inscribed polygon with the same number of sides.

The perimeters of two regular n-gons are related as a:b. How are the radii of their inscribed and circumscribed circles related?

How many sides does a regular polygon have, each of the internal angles of which is equal to: 1) 135; 2) 150?

Theorem 1. A circle can be circumscribed about any regular polygon.

Let ABCDEF (Fig. 419) be a regular polygon; it is necessary to prove that a circle can be circumscribed around it.

We know that it is always possible to draw a circle through three points that do not lie on the same line; hence, it is always possible to draw a circle that will pass through any three vertices of a regular polygon, for example, through the vertices E, D and C. Let the point O be the center of this circle.

Let us prove that this circle will also pass through the fourth vertex of the polygon, for example, through vertex B.

The segments OE, OD and OS are equal to each other, and each is equal to the radius of the circle. Let's draw another segment of the OB; it is impossible to immediately say about this segment that it is also equal to the radius of the circle, this must be proved. Consider triangles OED and ODC, they are isosceles and equal, therefore, ∠1 = ∠2 = ∠3 = ∠4.

If a inner corner given polygon is α , then ∠1 = ∠2 = ∠3 = ∠4 = α / 2 ; but if ∠4= α / 2 , then ∠5 = α / 2 , i.e. ∠4 = ∠5.

From this we conclude that (Delta)OSD = (Delta)OSV and, therefore, OB = OS, i.e., the segment OB is equal to the radius of the drawn circle. It follows from this that the circle will also pass through the vertex B of the regular polygon.

In the same way, we will prove that the constructed circle will pass through all the other vertices of the polygon. This means that this circle will be circumscribed about the given regular polygon. The theorem has been proven.

Theorem 2. A circle can be inscribed in any regular polygon.

Let ABCDEF be a regular polygon (Fig. 420), we must prove that a circle can be inscribed in it.

It is known from the previous theorem that a circle can be circumscribed near a regular polygon. Let point O be the center of this circle.

Connect the point O to the vertices of the polygon. The resulting triangles OED, ODC, etc. are equal to each other, which means that their heights drawn from the point O are also equal, i.e. OK = OL = OM = ON = OP = OQ.

Therefore, a circle circumscribed from the point O as from the center with a radius equal to the segment OK will pass through the points K, L, M, N, P and Q, and the heights of the triangles will be the radii of the circle. The sides of the polygon are perpendicular to the radii at those points, so they are tangent to that circle. And this means that the constructed circle is inscribed in the given regular polygon.

The same construction can be performed for any regular polygon, therefore, a circle can be inscribed in any regular polygon.

Consequence. A circle circumscribed about a regular polygon and inscribed in it has a common center.

Definitions.

1. The center of a regular polygon is the common center of the circles circumscribed about this polygon and inscribed in it.

2. The perpendicular dropped from the center of a regular polygon to its side is called the apothem of a regular polygon.

Expression of the sides of regular polygons in terms of the radius of the circumscribed circle

Via trigonometric functions one can express the side of any regular polygon in terms of the radius of the circle circumscribed about it.

Let AB be the side of the correct n-gon inscribed in a circle of radius OA = R (Fig.).

Let's apothemize the OD of a regular polygon and consider right triangle AOD. In this triangle

∠AOD = 1 / 2 ∠AOB = 1 / 2 360° / n= 180° / n

AD = AO sin ∠AOD = R sin 180° / n ;

but AB = 2AD and therefore AB = 2R sin 180° / n .

Correct side length n-gon inscribed in a circle is usually denoted a n, so the resulting formula can be written as follows:

a n= 2R sin 180° / n .

Consequences:

1. Side length of a regular hexagon inscribed in a circle of radius R , is expressed by the formula a 6=R, as

a 6 = 2R sin 180° / 6 = 2R sin 30° = 2R 1 / 2 = R.

2. Side length of a regular quadrilateral (square) inscribed in a circle of radius R , is expressed by the formula a 4 = R√2 , as

a 4 = 2R sin 180° / 4 = 2R sin 45° = 2R √ 2 / 2 = R√2

3. Side length of an equilateral triangle inscribed in a circle of radius R , is expressed by the formula a 3 = R√3 , as.

a 3 = 2R sin 180° / 3 = 2R sin 60° = 2R √ 3 / 2 = R√3

Area of ​​a regular polygon

Let the correct one be given n-gon (rice). It is required to determine its area. Denote the side of the polygon by a and the center through O. Connect the segments of the center with the ends of any side of the polygon, we get a triangle in which we draw the apothem of the polygon.

The area of ​​this triangle is Ah / 2. To determine the area of ​​the entire polygon, you need to multiply the area of ​​\u200b\u200bone triangle by the number of triangles, i.e. by n. We get: S = Ah / 2 n = ahn / 2 but an equals the perimeter of the polygon. Let's call it R.

Finally we get: S = P h / 2. where S is the area of ​​a regular polygon, P is its perimeter, h- apothem.

The area of ​​a regular polygon is equal to half the product of its perimeter and apothem.