Construction of regular polygons - technical drawing. Construction of regular polygons - technical drawing Convex 10-gon

Construction of a regular hexagon inscribed in a circle. The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to construct it, it is enough to divide the circle into six equal parts and connect the found points to each other (Fig. 60, a).

A regular hexagon can be built using a straight edge and a 30X60° square. To carry out this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4 (Fig. 60, b), construct sides 1 -6, 4-3, 4-5 and 7-2, after which we draw sides 5-6 and 3- 2.

Constructing an equilateral triangle inscribed in a circle. The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60° or just one compass.

Let's consider two ways to construct an equilateral triangle inscribed in a circle.

First way(Fig. 61,a) is based on the fact that all three angles of the triangle 7, 2, 3 contain 60°, and the vertical line drawn through point 7 is both the height and the bisector of angle 1. Since the angle is 0-1- 2 is equal to 30°, then to find the side

1-2, it is enough to construct an angle of 30° from point 1 and side 0-1. To do this, install the crossbar and square as shown in the figure, draw line 1-2, which will be one of the sides of the desired triangle. To construct side 2-3, set the crossbar in the position shown by the dashed lines, and draw a straight line through point 2, which will determine the third vertex of the triangle.

Second way is based on the fact that if you build a regular hexagon inscribed in a circle and then connect its vertices through one, you will get an equilateral triangle.

To construct a triangle (Fig. 61, b), mark the vertex-point 1 on the diameter and draw a diametrical line 1-4. Next, from point 4 with a radius equal to D/2, we describe an arc until it intersects with the circle at points 3 and 2. The resulting points will be the other two vertices of the desired triangle.

Constructing a square inscribed in a circle. This construction can be done using a square and a compass.

The first method is based on the fact that the diagonals of the square intersect in the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install the crossbar and square with angles of 45° as shown in Fig. 62, a, and mark points 1 and 3. Next, through these points we draw the horizontal sides of the square 4-1 and 3-2 using a crossbar. Then, using a straight edge, we draw the vertical sides of the square 1-2 and 4-3 along the leg of the square.

The second method is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter (Fig. 62, b). We mark points A, B and C at the ends of two mutually perpendicular diameters and from them with a radius y we describe arcs until they intersect each other.

Next, through the intersection points of the arcs we draw auxiliary straight lines, marked in the figure with solid lines. The points of their intersection with the circle will determine vertices 1 and 3; 4 and 2. We connect the vertices of the desired square obtained in this way in series with each other.

Construction of a regular pentagon inscribed in a circle.

To fit into a circle regular pentagon(Fig. 63), we make the following constructions.

We mark point 1 on the circle and take it as one of the vertices of the pentagon. We divide the segment AO in half. To do this, we describe an arc from point A with the radius AO until it intersects with the circle at points M and B. By connecting these points with a straight line, we get point K, which we then connect to point 1. With a radius equal to the segment A7, we describe an arc from point K until it intersects with the diametrical line AO ​​at point H. By connecting point 1 with point H, we get the side of the pentagon. Then, using a compass solution equal to the segment 1H, describing an arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass solution, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Constructing a regular pentagon along a given side.

To construct a regular pentagon along a given side (Fig. 64), we divide the segment AB into six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on line AB we draw a vertical line.

We get point 1-vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe an arc until it intersects with the arcs previously drawn from points A and B. The intersection points of the arcs determine the pentagon vertices 2 and 5. We connect the found vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Let a circle of diameter D be given; you need to fit a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of circle D, we describe an arc until it intersects with the continuation of the horizontal diameter at point F. We call point F the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / - // - /// from points IV, V and VI, draw horizontal lines until they intersect with the circle. We connect the found vertices sequentially to each other. A heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

The above method is suitable for constructing regular polygons with any number of sides.

The division of a circle into any number of equal parts can also be done using the data in Table. 2, which provides coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.

With ten angles and ten sides.

Regular decagon

A regular decagon has all sides of equal length and each interior angle measures 144°.

The area of ​​a regular decagon is equal to (t - side length):

$A\; =\; \backslash frac(5)(2)t^2\; \backslash cot\; \backslash frac(\backslash pi)(10)\; =\; \backslash frac(5t^2)(2)\; \backslash sqrt(5+2\backslash sqrt(5))\; \backslash approx\; 7.694\; t^2.$

Alternative formula $A=2.5dt$, where d is the distance between parallel sides or the diameter of the inscribed circle. IN trigonometric functions it is expressed like this:

$d=2t\backslash left(\backslash cos\backslash tfrac(3\backslash pi)(10)+\backslash cos\backslash tfrac(\backslash pi)(10)\backslash right),$

and can be represented in radicals as

$d=t\backslash sqrt(5+2\backslash sqrt(5)).$

The side of a regular decagon inscribed in a unit circle is $\backslash tfrac(\backslash sqrt(5)-1)(2)=\backslash tfrac(1)(\backslash varphi)$, Where $\backslash varphi$- golden ratio.

The radius of the circumscribed circle of the decagon is equal to

$R=\backslash frac(\backslash sqrt(5)+1)(2)t,$

and the radius of the inscribed circle is

$r=\backslash frac(\backslash sqrt(5+2\backslash sqrt(5)))(2)t.$

Construction

Otherwise it can be constructed as follows:

1. First construct a regular pentagon.
2. Connect all its vertices with the center of the circumcircle by straight lines until they intersect with the same circle on the opposite side. The remaining five vertices of the decagon are located at these intersection points.
3. Connect in order the vertices of the pentagon and the five points found a step earlier. The required decagon has been built.

Division of a regular decagon

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Links

By number of sides Correct Triangles Quadrilaterals See also

Polygons Star polygons Parquets on a plane Regular polyhedra and spherical parquets Kepler-Poinsot polyhedra Honeycomb Four-dimensional polyhedra

Excerpt describing the Decagon

-You're cold. You're shaking all over. You should go to bed,” she said.
- Go to bed? Yes, okay, I'll go to bed. “I’ll go to bed now,” Natasha said.
Since Natasha was told this morning that Prince Andrei was seriously wounded and was going with them, only in the first minute she asked a lot about where? How? Is he dangerously injured? and is she allowed to see him? But after she was told that she could not see him, that he was seriously wounded, but that his life was not in danger, she, obviously, did not believe what she was told, but was convinced that no matter how much she said, she would be answer the same thing, stopped asking and talking. All the way, with big eyes, which the countess knew so well and whose expression the countess was so afraid of, Natasha sat motionless in the corner of the carriage and now sat in the same way on the bench on which she sat down. She was planning something, something she was deciding or had already decided in her mind now - the countess knew this, but what it was, she did not know, and this frightened and tormented her.
- Natasha, undress, my dear, lie down on my bed. (Only the countess alone had a bed made on the bed; m me Schoss and both young ladies had to sleep on the floor on the hay.)
“No, mom, I’ll lie here on the floor,” Natasha said angrily, went to the window and opened it. The adjutant’s groan from the open window was heard more clearly. She stuck her head out into the damp air of the night, and the countess saw how her thin shoulders were shaking with sobs and beating against the frame. Natasha knew that it was not Prince Andrei who was moaning. She knew that Prince Andrei was lying in the same connection where they were, in another hut across the hallway; but this terrible incessant groan made her sob. The Countess exchanged glances with Sonya.
“Lie down, my dear, lie down, my friend,” said the countess, lightly touching Natasha’s shoulder with her hand. - Well, go to bed.
“Oh, yes... I’ll go to bed now,” said Natasha, hastily undressing and tearing off the strings of her skirts. Having taken off her dress and put on a jacket, she tucked her legs in, sat down on the bed prepared on the floor and, throwing her short thin braid over her shoulder, began to braid it. Thin, long, familiar fingers quickly, deftly took apart, braided, and tied the braid. Natasha's head turned with a habitual gesture, first in one direction, then in the other, but her eyes, feverishly open, looked straight and motionless. When the night suit was finished, Natasha quietly sank down onto the sheet laid on the hay on the edge of the door.
“Natasha, lie down in the middle,” said Sonya.
“No, I’m here,” Natasha said. “Go to bed,” she added with annoyance. And she buried her face in the pillow.
The Countess, m me Schoss and Sonya hastily undressed and lay down. One lamp remained in the room. But in the courtyard it was getting brighter from the fire of Malye Mytishchi two miles away, and the drunken cries of the people were buzzing in the tavern, which Mamon’s Cossacks had smashed, on the crossroads, on the street, and the incessant groan of the adjutant was heard.
Natasha listened for a long time to the internal and external sounds coming to her, and did not move. She heard first the prayer and sighs of her mother, the cracking of her bed under her, the familiar whistling snoring of m me Schoss, the quiet breathing of Sonya. Then the Countess called out to Natasha. Natasha did not answer her.
“He seems to be sleeping, mom,” Sonya answered quietly. The Countess, after being silent for a while, called out again, but no one answered her.
Soon after this, Natasha heard her mother's even breathing. Natasha did not move, despite the fact that her small bare foot, having escaped from under the blanket, was chilly on the bare floor.
As if celebrating victory over everyone, a cricket screamed in the crack. The rooster crowed far away, and loved ones responded. The screams died down in the tavern, only the same adjutant’s stand could be heard. Natasha stood up.
- Sonya? are you sleeping? Mother? – she whispered. Nobody answered. Natasha slowly and carefully stood up, crossed herself and stepped carefully with her narrow and flexible bare foot onto the dirty, cold floor. The floorboard creaked. She, quickly moving her feet, ran a few steps like a kitten and grabbed the cold door bracket.
It seemed to her that something heavy, striking evenly, was knocking on all the walls of the hut: it was beating her heart, frozen with fear, with horror and love, bursting.
She opened the door, crossed the threshold and stepped onto the damp, cold ground of the hallway. The gripping cold refreshed her. She felt the sleeping man with her bare foot, stepped over him and opened the door to the hut where Prince Andrei lay. It was dark in this hut. In the back corner of the bed, on which something was lying, there was a tallow candle on a bench that had burned out like a large mushroom.

A decagon, like all polygons, can be easily constructed using a compass and ruler. There are two simple ways to solve this interesting and unusual problem.

You will need

• - compass;
• - ruler.

Instructions

A polygon is a closed broken line. A decagon, accordingly, is a closed broken line consisting of 10 angles and 10 segments. Constructing an arbitrary decagon is not difficult. To do this, you need to take any 10 points that do not lie on the same line and connect these points with segments so that you get a closed figure. Moreover, the following condition must be met: any two points inside the resulting figure must be connected by a line that does not intersect the boundaries of the figure. If this condition is not met, then the constructed figure is not a polygon.

Method 1: Using a compass, draw a circle. Using a protractor, divide it into 10 equal sectors of 36 degrees each (360:10 = 36). Then connect sequentially all the points marked on the circle.

Method 2: Again, use a compass to draw a circle. Mark the center of the resulting circle with the letter O. Draw two perpendicular diameters of this circle CD and AB. Divide one of the 4 radii into two equal parts. The figure shows that the radius CO = SM + MO, where SM = MO.

Next, place the leg of the compass at point M and draw a circle with a radius equal to half the radius of the original circle. Using a ruler, connect the center of the small circle M with any of the 2 points (A or B) on the perpendicular diameter. In the figure, the center of a small circle is connected by stitch A. The length of the resulting segment AM will be equal to the length of the side of the decagon. All that remains is to make a compass opening equal to the length of the segment AM, place the leg of the compass at point A and mark the next point on the circle. Next, move the leg of the compass to a new point and mark the next one. And so on until 10 points equidistant from each other appear on the circle.