Regular pentagon: the necessary minimum of information. Regular pentagon

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 the horizontal line at point O. Finally, from point F we describe an arc that will intersect the circle at points H and K. Having plotted the distance FO = FH = FK along the circle five times and connecting the division points with lines, we get 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, we draw an arc of a circle of radius EM and mark with it the diameter AB at point F. The segment MF is equal to the side of 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 actual treated surfaces during painting work, 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, transporters 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 sensation in the world of mathematics. A new type of pentagons has been discovered that cover the plane without breaks and without overlaps.

This is only the 15th type of such pentagons and the first to be discovered in the last 30 years.

The plane is covered with triangles and quadrangles of any shape, but with pentagons everything is much more complicated and interesting. Regular pentagons cannot cover a plane, but some irregular pentagons can. The search for such figures has been one of the most interesting mathematical problems for a hundred years. The quest began in 1918, when mathematician Karl Reinhard discovered the first five suitable figures.

For a long time it was believed that Reinhard had calculated all possible formulas and that no more such pentagons existed, but in 1968 the mathematician R.B. Kershner found three more, and Richard James in 1975 brought their number to nine . That same year, 50-year-old American housewife and math enthusiast Marjorie Rice developed her own notation method and within a few years had discovered four more pentagons. Finally, in 1985, Rolf Stein increased the number of figures to fourteen.

Pentagons remain the only figure about which uncertainty and mystery remain. In 1963, it was proven that there are only three types of hexagons covering the plane. There are no such triangles among convex heptagonal, octagonal, and so on. But with the Pentagons, not everything is completely clear yet.

Until today, only 14 types of such pentagons were known. They are shown in the illustration. The formulas for each of them are given at the link.

For 30 years no one could find anything new, and finally the long-awaited discovery! It was made by a group of scientists from the University of Washington: Casey Mann, Jennifer McLoud and David Von Derau. This is what the little handsome guy looks like.

“We discovered the shape by computer searching through a large but limited number of variations,” says Casey Mann. - Of course, we are very excited and a little surprised that we were able to open new look pentagon."

The discovery seems purely abstract, but it may actually have practical applications. For example, in the production of finishing tiles.

The search for new pentagons covering the plane will certainly continue.

The pentagon represents geometric figure with five corners. 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

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

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°.

$\backslash frac((t^2\; \backslash sqrt\; (25\; +\; 10\backslash sqrt\; 5\; )\; ))(4)\; =$
$\backslash frac(5R^2)(4)\backslash sqrt(\backslash frac(5+\backslash sqrt(5$

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, a 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

Write a review about the article "Regular Pentagon"

Notes

Excerpt characterizing the Regular Pentagon

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 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, screams from different sides and more shots.
At the same moment 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 off 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 for 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.

Ozhegov’s explanatory dictionary states that a pentagon is a bounded by five intersecting straight lines forming five internal angles, as well as any object of a similar shape. If a given polygon has all the same sides and angles, then it is called regular (pentagon).

What is interesting about the regular pentagon?

It was in this form that the well-known building of the United States Department of Defense was built. From volumetric regular polyhedra only the dodecahedron has pentagon-shaped faces. And in nature there are absolutely no crystals whose faces would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of angles, which cannot be used to tile the area. Only a pentagon has the same number of diagonals as the number of sides. Agree, this is interesting!

Basic properties and formulas

Using formulas for an arbitrary regular polygon, you can determine all the necessary parameters that a pentagon has.

• Central angle α = 360 / n = 360/5 =72°.
• Internal angle β = 180° * (n-2)/n = 180° * 3/5 = 108°. Accordingly, the sum of the internal angles is 540°.
• The ratio of the diagonal to the side is (1+√5) /2, that is (approximately 1.618).
• The side length of a regular pentagon can be calculated using one of three formulas, depending on which parameter is already known:

• if a circle is described around it and its radius R is known, then a = 2*R*sin (α/2) = 2*R*sin(72°/2) ≈1.1756*R;
• in the case when a circle with radius r is inscribed in a regular pentagon, a = 2*r*tg(α/2) = 2*r*tg(α/2) ≈ 1.453*r;
• it happens that instead of radii, the value of the diagonal D is known, then the side is determined as follows: a ≈ D/1.618.

• The area of ​​a regular pentagon is determined, again, depending on what parameter we know:
• if there is an inscribed or circumscribed circle, then one of two formulas is used:

S = (n*a*r)/2 = 2.5*a*r or S = (n*R 2 *sin α)/2 ≈ 2.3776*R 2 ;

• The area can also be determined by knowing only the length of the side side a:

S = (5*a 2 *tg54°)/4 ≈ 1.7205* a 2 .

Regular pentagon: construction

This geometric figure can be constructed in different ways. For example, fit it into a circle with a given radius or build it on the basis of a given side. The sequence of actions was described in Euclid's Elements approximately 300 BC. In any case, we will need a compass and a ruler. Let's consider a method of construction using a given circle.

1. Select an arbitrary radius and draw a circle, marking its center with point O.

2. On the circle line, select a point that will serve as one of the vertices of our pentagon. Let this be point A. Connect points O and A with a straight line.

3. Draw a line through point O perpendicular to line OA. Designate the intersection of this straight line with the circle line as point B.

4. Halfway between points O and B, construct point C.

5. Now draw a circle whose center will be at point C and which will pass through point A. The place of its intersection with line OB (it will be inside the very first circle) will be point D.

6. Construct a circle passing through D, the center of which will be at A. The places of its intersection with the original circle should be marked with points E and F.

7. Now construct a circle whose center will be at E. This must be done so that it passes through A. Its other intersection point of the original circle must be marked

8. Finally, construct a circle through A with its center at point F. Label the other intersection of the original circle with point H.

9. Now all that remains is to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!