First way- on this side S using a protractor.
Draw a straight line and put AB = S on it; We take this line as a radius and use this radius to describe arcs from points A and B: then, using a protractor, we construct angles of 108° at these points, the sides of which will intersect with the arcs at points C and D; From these points with radius AB = 5 we describe arcs that intersect at E, and connect points L, C, E, D, B with straight lines.
The resulting pentagon- sought after.
Second way. Let's draw a circle of radius r. From point A, using a compass, draw an arc of radius AM until it intersects the circle at points B and C. We connect B and C with a line that intersects the horizontal axis at point E.
Then from point E we draw an arc that will intersect horizontal line at point O. Finally, from point F we describe an arc that will intersect the circle at points H and K. By plotting the distance FO = FH = FK along the circle five times and connecting the division points with lines, we obtain a regular pentagon.
Third way. Inscribe a regular pentagon in this circle. We draw two mutually perpendicular diameters AB and MC. Divide the radius AO by point E in half. From point E, as from the center, draw an arc of a circle of radius EM and mark with it the diameter AB at point F. Segment MF equal to side the desired regular pentagon. Using a compass solution equal to MF, we make serifs N 1, P 1, Q 1, K 1 and connect them with straight lines.
In the figure, a hexagon is constructed along this side.
Straight line AB = 5, as a radius, from points A and B we describe arcs that intersect at C; from this point, with the same radius, we describe a circle on which side A B will be deposited 6 times.
Hexagon ADEFGB- sought after.
"Designing rooms during renovation"
N.P. Krasnov
The basis for painting is the completely painted surfaces of walls, ceilings and other structures; painting is done using high-quality glue and oil paints, made for trimming or fluting. When starting to develop a finishing sketch, the master must clearly imagine the entire composition in a domestic environment and clearly understand the creative intent. Only if this basic condition is met can one correctly...
Measurement of work performed, with the exception of specially stated cases, is carried out based on the area of the actually treated surface, taking into account its relief and minus untreated areas. To determine the actually processed surfaces when painting works You should use the conversion factors given in the tables. A. Wooden window devices (measurement is made by the area of the openings along the outer contour of the frames) Name of devices Coefficient at ...
We have already said that to perform some types of painting work you need to be able to draw. And the ability to draw, in turn, presupposes knowledge of the rules for constructing geometric shapes. Sketches on paper are drawn using triangles, crossbars, transport and compasses, and on the plane of walls and ceilings constructions are made using a weight, ruler, wooden compass and cord. At the same time it is necessary...
A pentagon is a geometric figure with the appropriate number of angles. Moreover, for it, as for other types of polygons, there are general rules, including the sum of angles. A pentagon is a geometric figure with five angles. Moreover, from the point of view of geometry, the category of pentagons includes any polygons that have this characteristic, regardless of the location of its sides.
Sum of angles of a pentagon
A pentagon is actually a polygon, so to calculate the sum of its angles, you can use the formula adopted for calculating the specified sum in relation to a polygon with any number of angles. The above formula considers the sum of the angles of a polygon as the following equality: sum of angles = (n - 2) * 180°, where n is the number of angles in the desired polygon. Thus, in the case when we're talking about specifically about the pentagon, the value of n in this formula will be equal to 5. Thus, substituting the given value of n into the formula, it turns out that the sum of the angles of the pentagon will be 540°. However, it should be borne in mind that the application of this formula in relation to a specific pentagon is associated with a number of restrictions.Types of pentagons
The fact is that the indicated formula for a polygon with five angles, as well as for other types of these geometric figures, can only be applied if we are talking about a so-called convex polygon. It, in turn, is a geometric figure that satisfies next condition: all its points are on one side of the line that passes between two adjacent vertices. This definition can be simplified somewhat by noting that in this case geometric figure should not have vertices directed inside it. Only in this situation will the rule stating that the sum of the angles of a pentagon be 540° be true. One special case of a convex pentagon is a regular pentagon, all of whose angles are equal, each measuring 108 degrees. In geometry, it has a special name associated with its Greek root - pentagon. Thus, there is a whole category of pentagons, the sum of the angles in which will differ from the indicated value. So, for example, one of the variants of a non-convex pentagon is a star-shaped geometric figure. A star pentagon can also be obtained using the entire set of diagonals of a regular pentagon, that is, a pentagon: in this case, the resulting geometric figure will be called a pentagram, which has equal angles. In this case, the sum of the indicated angles will be 180°.A pentagon is a geometric figure with five angles. Moreover, from the point of view of geometry, the category of pentagons includes any polygons that have this characteristic, regardless of the location of its sides.
Sum of angles of a pentagon
A pentagon is actually a polygon, so to calculate the sum of its angles, you can use the formula adopted for calculating the specified sum in relation to a polygon with any number of angles. The above considers the sum of the angles of a polygon as the following equality: sum of angles = (n - 2) * 180°, where n is the number of angles in the desired polygon.Thus, in the case when we are talking specifically about , the value of n in this formula will be equal to 5. Thus, substituting the given value of n into the formula, it turns out that the sum of the angles of the pentagon will be 540°. However, it should be borne in mind that the application of this formula in relation to a specific pentagon is associated with a number of restrictions.
Types of pentagons
The fact is that the indicated formula, which has, as for other types of these geometric figures, can only be applied if we are talking about the so-called convex polygon. It, in turn, is a geometric figure that satisfies the following condition: all its points are on one side of the straight line that passes between two adjacent vertices.Thus, there is a whole category of pentagons, the sum of the angles in which will differ from the indicated value. So, for example, one of the variants of a non-convex pentagon is a star-shaped geometric figure. A star pentagon can also be obtained using the entire set of diagonals of a regular pentagon, that is, a pentagon: in this case, the resulting geometric figure will be called a pentagram, which has equal angles. In this case, the sum of the indicated angles will be 180°.
Polygon- a geometric figure on a plane, bounded by a closed broken line; a line that is obtained if we take n any points A 1, A 2, ..., A n and connect each of them with the next one, and the last one with the first one, with straight line segments.
There are two types of polygons: convex and non-convex. We'll take a closer look at convex polygons. A polygon is called convex, if no side of the polygon, being indefinitely extended, cuts the polygon into two parts. Convex polygons are regular and irregular, but we will consider the regular ones. Convex polygon called correct, if all sides are equal and all angles are equal. The center of a regular polygon is a point equidistant from all its vertices and all its sides.
The central angle of a regular polygon is the angle at which a side is visible from its center. Properties of a regular polygon:
1) A regular polygon is inscribed in a circle and circumscribed about the circle, with the centers of these circles coinciding;
2) The center of a regular polygon coincides with the centers of the inscribed and circumscribed circles;
3) Right side n-gon is related to the radius R circumscribed circle formula;
4) Perimeters are correct n-gons are related as the radii of circumscribed circles.
5) The diagonals of a regular n-gon divide its angles into equal parts.
Regular pentagon
Let's take a closer look at the regular pentagon - the pentagon.
Basic relationships: the angle at the vertex of a pentagon is 108°, external corner- 72°. The side of a pentagon is expressed in terms of the radii of the inscribed and circumscribed circle:
Let's construct a regular pentagon. This is easy to do using a circumscribed circle. From its center it is necessary to sequentially plot angles with the vertex in the center of the circle, equal to 72°. The sides of the corners will intersect the circle at five points, connecting them in series, we get a regular pentagon. Now let's draw all the diagonals in this pentagon. They form a regular star-shaped pentagon, i.e. the famous pentagram. It is interesting that the sides of the pentagrams, intersecting, again form a regular pentagon, in which the intersection of the diagonals gives us a new pentagram and so on ad infinitum (see Fig. 6).
A pentagram is a regular non-convex pentagon, it is also a regular stellated pentagon, or a regular pentagonal star. Many flowers, starfish and urchins, viruses, etc. have the shape of a five-pointed star. The first mention of the pentagram dates back to Ancient Greece. Translated from Greek, the pentagram literally means five lines. The pentagram was the hallmark of the Pythagorean school (580-500 BC). They believed that this beautiful polygon had many mystical properties. A reverent attitude towards the pentagram was also characteristic of medieval mystics, who borrowed a lot from the Pythagoreans. In the Middle Ages, it was believed that the pentagram served as a sign of protection from Satan.
Regular pentagon(Greek πενταγωνον ) - a geometric figure, a regular polygon with five sides.
Properties
- The dodecahedron is the only regular polyhedron whose faces are regular pentagons.
- The Pentagon, the building of the US Department of Defense, has the shape of a regular pentagon.
- A regular pentagon is a regular polygon with the fewest angles that cannot be tiled on a plane.
- In nature, there are no crystals with faces in the shape of a regular pentagon.
- The pentagon with all its diagonals is the projection of the 4-simplex.
see also
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Excerpt characterizing the Regular Pentagon
Petya didn’t know how long this lasted: he enjoyed himself, was constantly surprised by his pleasure and regretted that there was no one to tell it to. He was awakened by Likhachev's gentle voice.- Ready, your honor, you will split the guard in two.
Petya woke up.
- It’s already dawn, really, it’s dawning! - he screamed.
The previously invisible horses became visible up to their tails, and a watery light was visible through the bare branches. Petya shook himself, jumped up, took a ruble from his pocket and gave it to Likhachev, waved, tried the saber and put it in the sheath. The Cossacks untied the horses and tightened the girths.
“Here is the commander,” said Likhachev. Denisov came out of the guardhouse and, calling out to Petya, ordered them to get ready.
Quickly in the semi-darkness they dismantled the horses, tightened the girths and sorted out the teams. Denisov stood at the guardhouse, giving the last orders. The party's infantry, slapping a hundred feet, marched forward along the road and quickly disappeared between the trees in the predawn fog. Esaul ordered something to the Cossacks. Petya held his horse on the reins, impatiently awaiting the order to mount. Washed cold water, his face, especially his eyes, burned with fire, a chill ran down his back, and something in his whole body was trembling quickly and evenly.
- Well, is everything ready for you? - Denisov said. - Give us the horses.
The horses were brought in. Denisov became angry with the Cossack because the girths were weak, and, scolding him, sat down. Petya took hold of the stirrup. The horse, out of habit, wanted to bite his leg, but Petya, not feeling his weight, quickly jumped into the saddle and, looking back at the hussars who were moving behind in the darkness, rode up to Denisov.
- Vasily Fedorovich, will you entrust me with something? Please... for God's sake... - he said. Denisov seemed to have forgotten about Petya’s existence. He looked back at him.
“I ask you about one thing,” he said sternly, “to obey me and not to interfere anywhere.”
During the entire journey, Denisov did not speak a word to Petya and rode in silence. When we arrived at the edge of the forest, the field was noticeably getting lighter. Denisov spoke in a whisper with the esaul, and the Cossacks began to drive past Petya and Denisov. When they had all passed, Denisov started his horse and rode downhill. Sitting on their hindquarters and sliding, the horses descended with their riders into the ravine. Petya rode next to Denisov. The trembling throughout his body intensified. It became lighter and lighter, only the fog hid distant objects. Moving down and looking back, Denisov nodded his head to the Cossack standing next to him.
- Signal! - he said.
The Cossack raised his hand and a shot rang out. And at the same instant, the tramp of galloping horses was heard in front, shouts from different sides and more shots.
At the same instant as the first sounds of stomping and screaming were heard, Petya, hitting his horse and releasing the reins, not listening to Denisov, who was shouting at him, galloped forward. It seemed to Petya that it suddenly dawned as brightly as the middle of the day at that moment when the shot was heard. He galloped towards the bridge. Cossacks galloped along the road ahead. On the bridge he encountered a lagging Cossack and rode on. Some people ahead - they must have been French - were running with right side roads to the left. One fell into the mud under the feet of Petya's horse.
Cossacks crowded around one hut, doing something. A terrible scream was heard from the middle of the crowd. Petya galloped up to this crowd, and the first thing he saw was the pale face of a Frenchman with a shaking lower jaw, holding onto the shaft of a lance pointed at him.
“Hurray!.. Guys... ours...” Petya shouted and, giving the reins to the overheated horse, galloped forward down the street.
Shots were heard ahead. Cossacks, hussars and ragged Russian prisoners, running from both sides of the road, were all shouting something loudly and awkwardly. A handsome Frenchman, without a hat, with a red, frowning face, in a blue overcoat, fought off the hussars with a bayonet. When Petya galloped up, the Frenchman had already fallen. I was late again, Petya flashed in his head, and he galloped to where frequent shots were heard. Shots rang out in the courtyard of the manor house where he was with Dolokhov last night. The French sat down there behind a fence in a dense garden overgrown with bushes and fired at the Cossacks crowded at the gate. Approaching the gate, Petya, in the powder smoke, saw Dolokhov with a pale, greenish face, shouting something to the people. “Take a detour! Wait for the infantry!” - he shouted, while Petya drove up to him.
“Wait?.. Hurray!..” Petya shouted and, without hesitating a single minute, galloped to the place from where the shots were heard and where the powder smoke was thicker. A volley was heard, empty bullets squealed and hit something. The Cossacks and Dolokhov galloped after Petya through the gates of the house. The French, in the swaying thick smoke, some threw down their weapons and ran out of the bushes to meet the Cossacks, others ran downhill to the pond. Petya galloped on his horse along the manor's yard and, instead of holding the reins, strangely and quickly waved both arms and fell further and further out of the saddle to one side. The horse, running into the fire smoldering in the morning light, rested, and Petya fell heavily onto the wet ground. The Cossacks saw how quickly his arms and legs twitched, despite the fact that his head did not move. The bullet pierced his head.
After talking with the senior French officer, who came out to him from behind the house with a scarf on his sword and announced that they were surrendering, Dolokhov got off his horse and approached Petya, who was lying motionless, with his arms outstretched.
“Ready,” he said, frowning, and went through the gate to meet Denisov, who was coming towards him.
- Killed?! - Denisov cried out, seeing from afar the familiar, undoubtedly lifeless position in which Petya’s body lay.
“Ready,” Dolokhov repeated, as if pronouncing this word gave him pleasure, and quickly went to the prisoners, who were surrounded by dismounted Cossacks. - We won’t take it! – he shouted to Denisov.
