How to draw a regular pentagon. Regular pentagon: the required minimum of information
You cannot do without learning the technique of this process. There are several options for getting the job done. How to draw a star using a ruler will help you understand the most famous methods of this process.
Varieties of stars
There are many options for the appearance of a shape such as a star.
Since ancient times, its five-pointed variety has been used to draw pentagrams. This is due to its property, which allows you to make a drawing without lifting the pen from the paper.
There are also six-pointed, tailed comets.
The starfish traditionally has five peaks. Images of the Christmas version are often found of the same shape.
In any case, in order to draw a five-pointed star in stages, you need to resort to using special tools, since a freehand image is unlikely to look symmetrical and beautiful.
Executing a drawing
To understand how to draw an even star, you need to understand the essence of this figure.
The basis for its outline is a broken line, the ends of which converge at the starting point. It forms a regular pentagon - the pentagon.
The distinctive properties of such a figure are the ability to fit it into a circle, as well as a circle into this polygon.
All sides of the pentagon are equal. Understanding how to correctly execute a drawing, you can understand the essence of the process of constructing all the figures, as well as various schemes of parts and assemblies.
To achieve such a goal as to draw a star using a ruler, you need to have knowledge of the simplest mathematical formulas that are fundamental in geometry. You will also need the ability to count on a calculator. But the most important thing is logical thinking.
The job is not difficult, but it will require precision and meticulousness. The effort spent will be rewarded with a good symmetrical and therefore beautiful image of a five-pointed star.
Classical technique
The most famous way to draw a star using a compass, ruler and protractor is quite simple.
For this technique, you will need several tools: a compass or protractor, a ruler, a simple pencil, an eraser and a sheet of white paper.
To understand how to draw a star beautifully, you should act sequentially, stage by stage.
You can use special calculations in your work.
Figure calculation
At this stage of drawing the correct star, the contours of the finished shape appear.
If done correctly, the resulting image will be flat. This can be checked visually by rotating a sheet of paper and evaluating the shape. It will be the same at every turn.
The main contours are guided with a ruler and a simple pencil more clearly. All construction lines are removed.
To understand how to draw a star in stages, all actions should be carried out thoughtfully. In case of an error, you can correct the drawing with an eraser or carry out all the manipulations again.
Registration of work
The finished shape can be decorated in a variety of ways. The main thing is not to be afraid to experiment. Fantasy will suggest an original and beautiful image.
You can paint a drawn even star with a simple pencil or use a wide variety of colors and shades.
To figure out how to draw the right star, you need to stick to perfect lines throughout. Therefore, the most popular design option is to divide each ray of the shape into two equal parts with a line extending from the top to the center.
You don't have to separate the sides of the star with lines. It is allowed to simply paint over each ray of the figure with a darker shade from one side.
This option will also be the answer to the question of how to draw the correct star, because all its lines will be symmetrical.
If desired, with the aesthetic design of the figure, you can add an ornament or other various elements. By adding circles to the vertices, you can get a sheriff's star. By applying smooth feathering of the shadow sides, you can get a starfish.
This technique is the most common, as it effortlessly allows you to understand how to draw a five-pointed star in stages. Without resorting to complicated mathematical calculations, it is possible to get a correct, beautiful image.
After considering all the ways of how to draw a star using a ruler, you can choose the one that suits you best. The most popular is the geometric stepwise method. It is quite simple and effective. Applying fantasy and imagination, it is possible from the received correct, beautiful shape create an original composition. There are a great many design options for the picture. But you can always come up with your own, the most unusual and memorable plot. The main thing is not to be afraid to experiment!
This shape is a polygon with a minimum number of corners that cannot be paved with an area. Only the pentagon has the same number of diagonals as the number of its sides. Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has. For example, to inscribe it into a circle with a given radius, or build it on the basis of a given lateral side.
How to draw the beam correctly and what drawing accessories do you need? Take a piece of paper and mark a point in an arbitrary place. Then attach a ruler and draw a line from the indicated point to infinity. To draw a straight line, press the Shift key and draw a line of the desired length. Immediately after drawing, the Format tab opens. Remove the selection from the line and you will see a dot appear at the beginning of the line. To create a label, click the "Draw Label" button and create a field where the label will be located.
The first way to construct a pentagon is considered more "classic". The resulting shape will be a regular pentagon. The dodecagon is no exception, so its construction will be impossible without the use of a compass. The task of constructing a regular pentagon is reduced to the task of dividing a circle into five equal parts. You can draw a pentagram using the simplest tools.
I struggled for a long time trying to achieve this and independently find proportions and dependencies, but I did not succeed. It turned out that there are several different options for constructing a regular pentagon, developed by famous mathematicians. An interesting point is that this problem can be solved arithmetically only approximately accurately, since you will have to use irrational numbers. But it can be solved geometrically.
Division of circles. The points of intersection of these lines with the circle are the vertices of the square. A vertical diameter should be drawn in a circle of radius R (Step 1). At the conjugation point N of a straight line and a circle, the straight line is tangent to the circle.
Receiving with a strip of paper
A regular hexagon can be built using a rail and a 30X60 ° square. The vertices of such a triangle can be built using a compass and a square with angles of 30 and 60 °, or just one compass. To build side 2-3, set the raceway to the position shown by dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle. We mark point 1 on the circle and take it as one of the vertices of the pentagon. We connect the found vertices in series with each other. A heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.
And on the other end of the thread set a pencil and obsessed. If you know how to draw a star, but don't know how to draw a pentagon, draw a star with a pencil, then connect the adjacent ends of the star together, and then erase the star itself. Then put a sheet of paper (it is better to fix it on the table with four buttons or needles). Use buttons or needles to pin these 5 stripes to a piece of paper to keep them stationary. Then circle the resulting pentagon and remove these stripes from the sheet.
For example, we need to draw a five-pointed star (pentagram) for a picture of the Soviet past or the present of China. True, for this you need to be able to create a drawing of a star in perspective. Likewise, you can draw a shape with a pencil on paper. How to draw a star correctly, so that it looks smooth and beautiful, you cannot immediately answer.
From the center, lower 2 rays to the circumference, so that the angle between them is 72 degrees (protractor). The division of the circle into five parts is carried out using a conventional compass or protractor. Since the regular pentagon is one of the figures that contains the proportions of the golden ratio, painters and mathematicians have long been interested in its construction. These principles of construction with the use of a compass and a ruler were set forth in the Euclidean Principles.
If there is no compass at hand, then you can draw a simple star with five rays and then simply connect these rays. as you can see in the picture below, an absolutely regular pentagon is obtained.
Mathematics is a difficult science and it has many secrets of its own, some of which are quite amusing. If you are fond of such things, I advise you to find the book Funny Math.
A circle can be drawn not only with a compass. You can, for example, use a pencil and thread. We measure the required diameter on the thread. We clamp one end tightly on a sheet of paper, where we will draw a circle. And on the other end of the thread set a pencil and obsessed. Now it acts as with a compass: we pull the thread and around the circumference, slightly pressing with a pencil, draw a circle.
Draw peasants from the center inside the circle: a vertical line and a horizontal line. The intersection of the vertical line and the circle will be the vertex of the pentagon (point 1). Now divide the right half of the horizontal line in half (point 2). We measure the distance from this point to the vertex of the pentagon and this segment is laid to the left of point 2 (point 3). Using a thread and a pencil, draw an arc from point 1 with a radius to point 3, intersecting the first circle on the left and right - the intersection points will be the vertices of the pentagon. Let's designate their points 4 and 5.
Now from point 4 we make an arc that intersects the circle at the bottom, with a radius equal to the length from point 1 to 4 - this will be point 6. In the same way, from point 5 - we will denote it by point 7.
It remains to connect our pentagon with vertices 1, 5, 7, 6, 4.
I know how to build a simple pentagon using a compass: Draw a circle, mark five points, connect them. You can build a pentagon with equal sides, for this we still need a protractor. We just put the same 5 points along the protractor. To do this, mark the angles of 72 degrees. After that, we also connect with segments and get the shape we need.
The green circle can be drawn with an arbitrary radius. We will inscribe a regular pentagon into this circle. It is impossible to draw a circle exactly without a compass, but it is not necessary. The circle and all further construction can be done by hand. Next, through the center of the circle O, you need to draw two mutually perpendicular straight lines and one of the intersection points of the straight line with the circle, denote A. Point A will be the vertex of the pentagon. We divide the radius of the OB in half and put the point C. From the point C we draw the second circle with the radius AC. From point A we draw a third circle with radius AD. The points of intersection of the third circle with the first (E and F) will also be the vertices of the pentagon. From points E and F with radius AE we make serifs on the first circle and get the remaining vertices of the pentagon G and H.
Adepts of black art: in order to draw a pentagon simply, beautifully and quickly, you should draw a correct, harmonious basis for the pentagram (five-pointed star) and connect the ends of the rays of this star by means of straight, even lines. If everything was done correctly, the connecting line around the base will be the desired pentagon.
(in the picture - a completed but unfilled pentagram)
For those who are unsure of the correctness of the pentagram: take the Vitruvian man Da Vinci as a basis (see below)
If you need a pentagon - poke randomly 5 points and their outer contour will be a pentagon.
If you need a regular pentagon, then without a mathematical compass, this construction is impossible, since without it it is impossible to draw two identical but not parallel segments. Any other tool that allows you to draw two identical, but not parallel lines is equivalent to a mathematical compass.
First you need to draw a circle, then the guides, then the second dotted circle, find the top point, then measure the two upper corners, draw the lower ones from them. Note that the radius of the compass is the same for the entire construction.
It all depends on which pentagon you need. If any, then put five points and connect them to each other (naturally, we do not put the points in a straight line). And if you need a pentagon of the correct shape, take any five in length (strips of paper, matches, pencils, etc.), lay out the pentagon and outline it.
A pentagon can be drawn, for example, from a star. If you know how to draw a star, but don't know how to draw a pentagon, draw a star with a pencil, then connect the adjacent ends of the star together, and then erase the star itself.
Second way. Cut a strip of paper with a length equal to the desired side of the pentagon and a narrow width, say 0.5 - 1 cm.As a template, cut four more of the same stripes along this strip to make a total of 5.
Then put a sheet of paper (it is better to fix it on the table with four buttons or needles). Then place these 5 stripes on a piece of paper so that they form a pentagon. Use buttons or needles to pin these 5 stripes to a piece of paper to keep them stationary. Then circle the resulting pentagon and remove these stripes from the sheet.
If there is no compass and you need to build a pentagon, then I can advise the following. I built it myself. You can draw a regular five-pointed star. And after that, to get a pentagon, you just need to connect all the vertices of the star. This is how the pentagon will turn out. Here's what we get
With smooth black lines, we connected the vertices of the star and got a pentagon.
5.3. Golden Pentagon; construction of Euclid.
A wonderful example of the "golden ratio" is a regular pentagon - convex and star-shaped (Fig. 5).
To build a pentagram, you need to build a regular pentagon.
Let O be the center of the circle, A be a point on the circle, and E the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, we postpone the segment CE \u003d ED on the diameter. The side length of a regular pentagon inscribed in a circle is DC. We put aside the segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All the diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36 ° at the apex, and the base, laid on side, divides it in the proportion of the golden ratio.
There is also a golden cuboid - this rectangular parallelepiped with ribs having lengths 1.618, 1 and 0.618.
Now consider the proof proposed by Euclid in the "Elements".
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from the center of the circumscribed circle. Let's start with
segment ABE, divided on the average and
So, let AC \u003d AE. We denote by a equal angles EMU and SEB. Since AC \u003d AE, the ACE angle is also equal to a. The theorem that the sum of the angles of a triangle is 180 degrees, allows you to find the angle ALL: it is 180-2a, and the angle EAC - 3a - 180. But then the angle ABC is 180-a. Summing up the angles of the triangle ABC, we obtain
180 \u003d (3a -180) + (3a-180) + (180 - a)
Whence 5a \u003d 360, then a \u003d 72.
So, each of the angles at the base of the triangle is WEIGHT twice the apex angle of 36 degrees. Therefore, in order to build a regular pentagon, it is only necessary to draw any circle with center at point E, intersecting EC at point X and side EB at point Y: the segment XY serves as one of the sides of the regular pentagon inscribed in the circle; By going around the entire circle, you can find all the other sides.
Let us now prove that AC \u003d AE. Suppose that the vertex C is connected by a straight line segment with the midpoint N of the segment BE. Note that since CB \u003d CE, the angle СNЕ is straight. By the Pythagorean theorem:
CN 2 \u003d a 2 - (a / 2j) 2 \u003d a 2 (1-4j 2)
Hence we have (AC / a) 2 \u003d (1 + 1 / 2j) 2 + (1-1 / 4j 2) \u003d 2 + 1 / j \u003d 1 + j \u003d j 2
So, AC \u003d jа \u003d jAB \u003d AE, as required to prove
5.4 The spiral of Archimedes.
Sequentially cutting off the squares from the golden rectangles to infinity, each time connecting opposite points with a quarter circle, we get a rather graceful curve. The first to pay attention to her was the ancient Greek scientist Archimedes, whose name she bears. He studied it and derived the equation for this spiral.
Currently, the Archimedes spiral is widely used in technology.
6. Fibonacci numbers.
The name of the Italian mathematician Leonardo from Pisa, who is better known by his nickname Fibonacci (Fibonacci is abbreviated filius Bonacci, that is, the son of Bonacci), is indirectly associated with the golden ratio.
In 1202. he wrote the book "Liber abacci", that is, "The Book of the Abacus". "Liber abacci" is a voluminous work containing almost all the arithmetic and algebraic information of that time and played a significant role in the development of mathematics in Western Europe over the next several centuries. In particular, it was through this book that Europeans got acquainted with Hindu ("Arabic") numbers.
The material reported in the book is explained on a large number of problems that make up a significant part of this treatise.
Consider one such problem:
"How many pairs of rabbits are born from one pair in one year?
Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, in order to find out how many pairs of rabbits will be born during this year, if the nature of rabbits is such that in a month a pair of rabbits will reproduce another, and rabbits give birth from the second month after their birth "
| Months | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Pair of rabbits | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 |
Let's move on from rabbits to numbers and consider the following number sequence:
u 1, u 2 ... u n
in which each term is equal to the sum of the previous two, i.e. for any n\u003e 2
u n \u003d u n -1 + u n -2.
This sequence asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.
If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13: 8), the result will be a value fluctuating around the irrational value of 1.61803398875 ... and one time it surpasses or does not reach it.
The asymptotic behavior of a sequence, damping fluctuations of its ratio around an irrational number Ф can become clearer if we show the ratios of several first terms of the sequence. This example shows the relationship of the second term to the first, the third to the second, the fourth to the third, and so on:
1: 1 \u003d 1.0000, which is 0.6180 less phi
2: 1 \u003d 2.0000, which is 0.3820 more phi
3: 2 \u003d 1.5000, which is 0.1180 less phi
5: 3 \u003d 1.6667, which is 0.0486 more phi
8: 5 \u003d 1.6000, which is 0.0180 less phi
As we move along the Fibonacci summation sequence, each new term will divide the next one with more and more approximation to the unattainable F.
A person subconsciously seeks divine proportion: it is needed to satisfy his need for comfort.
When dividing any member of the Fibonacci sequence by the next one, you get just the opposite to 1.618 value (1: 1.618 \u003d 0.618). But this is also a very unusual, even remarkable phenomenon. Since the original relationship is infinite, there should be no end to this relationship either.
When dividing each number by the next one after it, we get the number 0.382
By choosing the ratios in this way, we get the main set of Fibonacci ratios: 4.235, 2.618, 1.618,0.618,0.382,0.236. Let us also mention 0.5. All of them play a special role in nature and, in particular, in technical analysis.
It should be noted here that Fibonacci only reminded humanity of his sequence, since it was known back in ancient times called the Golden Ratio.
The golden ratio, as we have seen, arises in connection with the regular pentagon, therefore the Fibonacci numbers play a role in everything that is related to regular pentagons - convex and star-shaped.
The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of golden division. Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich, using Fibonacci numbers, solves Hilbert's 10th problem (on the solution of Diophantine equations). There are sophisticated methods for solving a number of cybernetic problems (search theory, games, programming) using the Fibonacci numbers and the golden ratio. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.
One of the advances in this area is the discovery of generalized Fibonacci numbers and generalized golden ratios. Fibonacci series (1, 1, 2, 3, 5, 8) and the "binary" series of numbers 1, 2, 4, 8, 16 ... (that is, a series of numbers up to n, where any natural number less than n can be represented as the sum of some numbers in this series) at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 \u003d 1 + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... Is it possible to find a general mathematical formula from which and " binary "series, and the Fibonacci series?
Indeed, let us set a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5 ... Consider number series, S + 1 of the first members of which are units, and each of the subsequent ones is equal to the sum of two members of the previous one and the one S steps away from the previous one. If nth term of this series we denote by S (n), then we obtain general formula S (n) \u003d S (n - 1) + S (n - S - 1).
Obviously, for S \u003d 0 from this formula we get a "binary" series, for S \u003d 1 - the Fibonacci series, with S \u003d 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.
In general, the golden S-proportion is the positive root of the golden S-section equation x S + 1 - x S - 1 \u003d 0.
It is easy to show that when S \u003d 0, the segment is divided in half, and when S \u003d 1, the familiar classical golden ratio.
The ratios of neighboring Fibonacci S-numbers coincide with absolute mathematical precision in the limit with the golden S-proportions! That is, the golden S-ratios are the numerical invariants of the Fibonacci S-numbers.
7.The Golden Ratio in Art.
7.1. The golden ratio in painting.
Moving on to examples of the "golden ratio" in painting, one cannot help but focus on the work of Leonardo da Vinci. His personality is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one, not being a mathematician, dare to read my works."
There is no doubt that Leonardo da Vinci was a great artist, this was already recognized by his contemporaries, but his personality and activity will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “about everything in the world. "
The portrait of Monna Lisa (La Gioconda) has attracted the attention of researchers for many years, who discovered that the composition of the drawing is based on golden triangles, which are parts of a regular star-shaped pentagon ..
Also, the proportion of the golden ratio appears in Shishkin's painting. In this famous painting by I.I. Shishkin, the motives of the golden section are clearly visible. A pine tree brightly lit by the sun (in the foreground) divides the length of the painting along the golden ratio. To the right of the pine is a sunlit hillock. He divides the right side of the picture horizontally along the golden ratio.
In Raphael's painting "The Beating of Babies", another element of the golden ratio is visible - the golden spiral. On the preparatory sketch of Raphael, red lines are drawn from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman holding him close to her, the warrior with the sword drawn and then along the figures of the same group on the right side of the sketch ... It is unknown if Raphael built or felt the golden spiral.
T. Cook used the golden section in his analysis of the painting by Sandro Botticelli "The Birth of Venus".
7.2. The pyramids of the golden ratio.
The medical properties of the pyramids, especially the golden ratio, are widely known. According to some of the most common opinions, the room in which such a pyramid is located appears larger and the air is more transparent. Dreams start to be remembered better. It is also known that the golden ratio was widely used in architecture and sculpture. An example of this was: the Pantheon and Parthenon in Greece, the buildings of architects Bazhenov and Malevich
8. Conclusion.
It must be said that the golden ratio has great application in our life.
It has been proven that the human body is divided in the proportion of the golden ratio by the belt line.
The shell of the nautilus is twisted like a golden spiral.
Thanks to the golden ratio, the asteroid belt was discovered between Mars and Jupiter - according to the proportion there should be another planet.
Excitation of the string at the point dividing it in relation to the golden division will not cause the string to vibrate, that is, it is a compensation point.
On aircraft with electromagnetic energy sources, rectangular cells with the proportion of the golden ratio are created.
La Gioconda is built on golden triangles, a golden spiral is present in Raphael's painting "The Beating of Babies".
The proportion found in the painting by Sandro Botticelli "The Birth of Venus"
There are many architectural monuments built using the golden ratio, including the Pantheon and Parthenon in Athens, the buildings of architects Bazhenov and Malevich.
John Kepler, who lived five centuries ago, said: "Geometry has two great treasures. The first is the Pythagorean theorem, the second is the division of a segment in the extreme and mean ratio"
List of references
1.D. Pidow. Geometry and Art. - M .: Mir, 1979.
2. Journal "Science and Technology"
3. Journal "Quant", 1973, No. 8.
4. Journal "Mathematics in School", 1994, No. 2; No. 3.
5. Kovalev F.V. The golden ratio in painting. K .: Vyscha school, 1989.
6. Stakhov A. Codes of the golden ratio.
7.Vorobyov N.N. "Fibonacci numbers" - Moscow: Science 1964
8. "Mathematics - Encyclopedia for Children" M .: Avanta +, 1998
9. Information from the Internet.
Fibonacci matrices and the so-called "golden" matrices, new computer arithmetic, new coding theory and new theory cryptography. The essence of the new science, in the revision from the point of view of the golden ratio of all mathematics, starting with Pythagoras, which, naturally, will entail new and probably very interesting mathematical results in the theory. In practical terms - "golden" computerization. And since ...
Will not affect this result. The base of the golden ratio is an invariant of recursive relations 4 and 6. This is the “stability” of the golden ratio, one of the principles of the organization of living matter. Also, the base of the golden ratio is the solution of two exotic recursive sequences (Fig. 4.) 4 Recursive Fibonacci Sequences So ...
The ear is j5 and the distance from ear to crown is j6. Thus, in this statue we see a geometric progression with the denominator j: 1, j, j2, j3, j4, j5, j6. (fig. 9). Thus, the golden ratio is one of the fundamental principles in the art of ancient Greece. Rhythms of the heart and brain. The human heart beats evenly - about 60 beats per minute at rest. The heart squeezes like a piston ...
Ozhegov's Explanatory Dictionary says that a pentagon is bounded by five intersecting straight lines forming five inner corners, as well as any object of a similar shape. If a given polygon has all sides and angles the same, 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. Of the volumetric regular polyhedra, only the dodecahedron has pentagon-shaped faces. And in nature, crystals are completely absent, the faces of which would resemble a regular pentagon. In addition, this shape is a polygon with a minimum number of corners that cannot be paved with an area. Only the pentagon has the same number of diagonals as the number of its sides. Agree, this is interesting!
Basic properties and formulas
Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has.
- Central angle α \u003d 360 / n \u003d 360/5 \u003d 72 °.
- Internal angle β \u003d 180 ° * (n-2) / n \u003d 180 ° * 3/5 \u003d 108 °. Accordingly, the sum of the interior angles is 540 °.
- The ratio of the diagonal to the side is (1 + √5) / 2, which is (approximately 1.618).
- The length of the side that a regular pentagon has 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 \u003d 2 * R * sin (α / 2) \u003d 2 * R * sin (72 ° / 2) ≈1,1756 * R;
- in the case when a circle with radius r is inscribed in a regular pentagon, a \u003d 2 * r * tan (α / 2) \u003d 2 * r * tan (α / 2) ≈ 1.453 * r;
- it happens that instead of the radii, the value of the diagonal D is known, then the side is determined as follows: a ≈ D / 1.618.
- The area of \u200b\u200ba regular pentagon is determined, again, depending on which parameter we know:
- if there is an inscribed or circumscribed circle, then one of two formulas is used:
S \u003d (n * a * r) / 2 \u003d 2.5 * a * r or S \u003d (n * R 2 * sin α) / 2 ≈ 2.3776 * R 2;
- the area can also be determined by knowing only the length of the lateral side a:
S \u003d (5 * a 2 * tg54 °) / 4 ≈ 1.7205 * a 2.
Regular pentagon: construction
This geometric shape can be built in different ways. For example, to inscribe it into a circle with a given radius, or build it on the basis of a given lateral side. The sequence of actions was described as early as in the "Beginnings" of Euclid about 300 BC. In any case, we need a compass and a ruler. Consider a construction method using a given circle.
1. Select an arbitrary radius and draw a circle, marking its center with point O.
2. On the line of the circle, select a point that will serve as one of the vertices of our pentagon. Let it be point A. Connect points O and A with a straight line.
3. Draw a line through point O perpendicular to line OA. The place of intersection of this straight line with the circle line is designated as point B.
4. In the middle of the distance between points O and B, draw point C.
5. Now draw a circle, the center of which 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. Places of its intersection with the original circle must be designated by points E and F.
7. Now draw a circle, the center of which will be in E. This must be done so that it passes through A. Its other place of intersection of the original circle must be designated
8. Finally, draw a circle through A with center at F. Mark the other intersection of the original circle with point H.
9. Now it remains only to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!
