Draw Pentagon in Circle Without Using Any Measures

Positive pentagon is a polygon in which all five sides and all five angles are equal. Information technology is easy to describe a circle around it. Erect pentagon and this circle will aid.

Instruction

1. First of all, you demand to build a circle with a compass. Let the eye of the circle coincide with indicate O. Draw axes of symmetry perpendicular to each other. At the intersection point of 1 of these axes with the circle, put a indicate 5. This point volition be the meridian of the future pentagon but. Place bespeak D at the bespeak of intersection of some other centrality with the circle.

2. On the segment OD, find the middle and mark indicate A in it. Later, you need to depict a circumvolve with a compass centered at this point. In addition, it must laissez passer through the point V, that is, with radius CV. Designate the point of intersection of the axis of symmetry and this circumvolve every bit B.

three. After, with the assist compass draw a circle of the same radius, placing the needle at indicate V. Designate the intersection of this circle with the original one every bit point F. This indicate will become the 2nd vertex of the future truthful pentagon but.

4. Now it is necessary to describe the same circle through point Due east, simply with the eye at F. Designate the intersection of the circle just drawn with the original one as bespeak G. This point will also go i of the vertices pentagon just. Similarly, you need to build another circle. Its center is in G. Let information technology intersect with the original circle H. This is the last vertex of a true polygon.

five. You should have five vertices. It remains piece of cake to combine them along the line. As a result of all these operations, you will go a positive inscribed in a circle. pentagon .

Building positive pentagons immune with the support of a compass and straightedge. True, the process is rather long, as, however, is the structure of any positive polygon with an odd number of sides. Modernistic computer programs allow yous to exercise this in a few seconds.

Yous will need

• - A reckoner with AutoCAD software.

Teaching

1. Find the superlative menu in the AutoCAD program, and in it the "Basic" tab. Click on it with the left mouse push button. The Draw console appears. Various types of lines will announced. Select a closed polyline. It is a polygon, information technology remains only to enter the parameters. AutoCAD. Allows you to depict a diversity of regular polygons. The number of sides tin can be up to 1024. You tin also use the command line, depending on the version, by typing "_polygon" or "multi-bending".

2. Regardless of whether yous utilize the command line or context menus, you will encounter a window on the screen in which you are prompted to enter the number of sides. Enter the number "five" at that place and press Enter. You will be prompted to decide the centre of the pentagon. Enter the coordinates in the box that appears. It is allowed to denote them as (0,0), only there may exist any other data.

3. Select the required construction method. . AutoCAD offers three options. A pentagon tin can be described around a circle or inscribed in it, but it is also immune to build it according to a given side size. Select the desired option and printing enter. If necessary, set the radius of the circumvolve and also press enter.

4. A pentagon on a given side is first constructed correctly in the same way. Select Depict, a closed polyline, and enter the number of sides. Right-click to open the context menu. Press the command "border" or "side". In the control line, blazon the coordinates of the initial and final points of ane of the sides of the pentagon. Subsequently this pentagon will appear on the screen.

5. All operations can be performed with command line support. Say, to build a pentagon along the side in the Russian version of the program, enter the letter of the alphabet "c". In the English version it will be "_e". In guild to build an inscribed or circumscribed pentagon, enter later on the number of sides of the letter "o" or "c" (or the English language "_s" or "_i")

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Useful advice
With such a elementary method, it is possible to build not only a pentagon. In order to construct a triangle, yous need to spread the legs of the compass to a altitude equal to the radius of the circumvolve. After that, place the needle at any point. Depict a thin auxiliary circle. 2 points of intersection of the circles, as well as the point where the leg of the compass was, grade three vertices of a positive triangle.

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

A regular hexagon can exist constructed using a T-square and a 30X60° square. To perform this construction, nosotros take the horizontal diameter of the circle equally the bisector of angles i and four (Fig. 60, b), build sides 1-6, 4-3, 4-5 and seven-2, after which nosotros depict sides v-six and iii- 2.

Structure of an equilateral triangle inscribed in a circle. The vertices of such a triangle can be synthetic using a compass and a square with angles of 30 and threescore °, or only 1 compass.

Consider ii ways to construct an equilateral triangle inscribed in a circle.

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

1-two, it is plenty to build an angle of xxx ° at point i and side 0-one. To practice this, set the T-square and foursquare as shown in the figure, draw a line one-2, which will exist ane of the sides of the desired triangle. To build side 2-3, set the T-square to the position shown by the dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle.

2d style based on the fact that if you build regular hexagon, inscribed in a circle, and then connect its vertices through one, you get an equilateral triangle.

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

Structure of a foursquare inscribed in a circle. This structure 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 confining circle and are inclined to its axes at an angle of 45°. Based on this, nosotros install a T-square and a foursquare with angles of 45 ° as shown in Fig. 62, a, and mark points 1 and 3. Further, through these points, we draw the horizontal sides of the foursquare 4-1 and three-2 with the help of a T-foursquare. Then, using a T-square along the leg of the square, we describe the vertical sides of the foursquare 1-2 and 4-iii.

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 the arcs until they intersect.

Further, through the points of intersection of the arcs, we depict auxiliary lines, marked on the figure with solid lines. Their points of intersection with the circle will ascertain vertices ane and iii; iv and 2. The vertices of the desired foursquare obtained in this way are connected in series with each other.

Structure of a regular pentagon inscribed in a circle.

To inscribe a regular pentagon in a circumvolve (Fig. 63), we make the following constructions.

We marker indicate 1 on the circle and take it equally one of the vertices of the pentagon. Split segment AO in half. To do this, with the radius AO from betoken A, we describe the arc until it intersects with the circle at points K and B. Connecting these points with a straight line, we get point Grand, which we and then connect with indicate 1. With a radius equal to segment A7, nosotros draw the arc from bespeak Thousand to the intersection with the diametrical line AO ​​at point H. Connecting betoken 1 with point H, nosotros go the side of the pentagon. Then, with a compass opening equal to the segment 1H, having described the arc from vertex i to the intersection with the circle, nosotros find vertices two and 5. Having fabricated serifs from vertices 2 and five with the same compass opening, we obtain the remaining vertices 3 and four. We connect the found points sequentially with each other.

Construction of a regular pentagon given its side.

To construct a regular pentagon along its given side (Fig. 64), we divide the segment AB into 6 equal parts. From points A and B with radius AB we depict arcs, the intersection of which will give signal 1000. Through this point and sectionalization three on the line AB we draw a vertical line.

Nosotros go the point 1-vertex of the pentagon. Then, with a radius equal to AB, from betoken i we describe the arc to the intersection with the arcs previously drawn from points A and B. The intersection points of the arcs determine the vertices of the pentagon two and v. We connect the establish vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Allow a circle of diameter D be given; you need to inscribe a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circumvolve into seven equal parts. From point 7 with a radius equal to the bore of the circle D, we describe the arc until information technology intersects with the continuation of the horizontal diameter at betoken F. Point F is chosen the pole of the polygon. Taking point VII as one of the vertices of the heptagon, nosotros depict rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circumvolve will determine the vertices VI, V and 4 of the heptagon. To obtain vertices / - // - /// from points IV, Five and 6, nosotros describe horizontal lines until they intersect with the circle. We connect the found vertices in series with each other. The heptagon tin can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

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

The division of a circle into whatever number of equal parts can also be done using the information in Table. 2, which shows the coefficients that go far possible to determine the dimensions of the sides of regular inscribed polygons.

5.3. golden pentagon; construction of Euclid.

Corking example The "golden section" is a regular pentagon - convex and stellate (Fig. 5).

To build a pentagram, yous need to build a regular pentagon.

Let O be the center of the circle, A a point on the circumvolve, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at signal O, intersects with the circumvolve at point D. Using a compass, marking the segment CE = ED on the diameter. The length of a side of a regular pentagon inscribed in a circle is DC. Nosotros set aside segments DC on the circle and go v points for cartoon a regular pentagon. We connect the corners of the pentagon through i diagonal and get a pentagram. All 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 top, and the base of operations laid on the side divides it in proportion to the golden section.

There is also a golden cuboid - this is a rectangular parallelepiped with edges having lengths of 1.618, one and 0.618.

At present consider the proof offered by Euclid in the Elements.

Permit us at present encounter how Euclid uses golden ratio in order to build an bending of 72 degrees - it is at this angle that the side of a regular pentagon is visible

from the center of the confining circle. Permit'southward start with

segment ABE, divided in the middle and

So let Air-conditioning = AE. Denote by a equal angles EMU and SEV. Since Air-conditioning=AE, the angle ACE 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 is 3a - 180. But so the angle ABC is 180-a. Summing upwardly the angles of triangle ABC, we get

180=(3a -180) + (3a-180) + (180 - a)

Whence 5a=360, so a=72.

So, each of the angles at the base of operations of the triangle BEC is twice the angle at the top, equal to 36 degrees. Therefore, in order to construct a regular pentagon, it is only necessary to draw any circumvolve centered at point E, intersecting EC at point 10 and the side EB at betoken Y: the segment XY is one of the sides of the regular pentagon inscribed in the circle; Going effectually the entire circle, yous tin find all the other sides.

We now prove that Ac=AE. Suppose that the vertex C is connected past a directly line segment to the midpoint North of the segment Be. Annotation that since CB = CE, so the angle CNE is a correct angle. Co-ordinate to the Pythagorean theorem:

CN 2 \u003d a 2 - (a / 2j) 2 \u003d a two (1-4j 2)

Hence nosotros have (AC/a) 2 = (1+one/2j) ii + (1-i/4j 2) = 2+1/j = one + j =j two

So, AC = ja = jAB = AE, which was to be proved

5.4. Spiral of Archimedes.

Successively cutting off squares from the golden rectangles to infinity, each time connecting the opposite points with a quarter of a circumvolve, nosotros go a rather elegant curve. The first attention was fatigued to her by the ancient Greek scientist Archimedes, whose proper name she bears. He studied information technology and deduced the equation of this spiral.

Currently, the Archimedes screw is widely used in technology.

half-dozen. Fibonacci numbers.

The name of the Italian mathematician Leonardo from Pisa, who is amend known past his nickname Fibonacci (Fibonacci is an abridgement of filius Bonacci, that is, the son of Bonacci), is indirectly associated with the aureate ratio.

In 1202 he wrote the book "Liber abacci", that is, "The Volume of the abacus". "Liber abacci" is a voluminous work containing about all the arithmetic and algebraic information of that fourth dimension and played a pregnant role in the development of mathematics in Western Europe over the next few centuries. In detail, information technology was from this book that Europeans became acquainted with Hindu ("Standard arabic") numerals.

The material presented in the book is explained in large numbers problems that make up a significant part of this treatise.

Consider one such problem:

How many pairs of rabbits are built-in from 1 pair in one yr?

Someone placed a pair of rabbits in a sure place, enclosed on all sides by a wall, in club to discover 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 some other, and rabbits give nascence from the second month afterward their nascence "

Months	1	2	3	4	5	6	7	8	9	10	11	12
Pairs of rabbits	2	3	5	viii	xiii	21	34	55	89	144	233	377

Let united states now move from rabbits to numbers and consider the post-obit number sequence:

u i , u two … u north

in which each term is equal to the sum of the two previous ones, i.e. for any n>ii

u n \u003d u northward -ane + u n -2.

This sequence asymptotically (approaching more and more slowly) tends to some constant relation. All the same, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional office. It cannot be expressed exactly.

If any member of the Fibonacci sequence is divided by the one preceding it (for case, thirteen:viii), the upshot will be a value fluctuating around the irrational value 1.61803398875... and either exceeding it or not reaching it every other time.

Asymptotic behavior of the sequence, damped oscillations its ratio is about irrational numberФ can become more understandable if you show the relationship of several first terms of the sequence. This case shows the human relationship of the 2nd term to the outset, the third to the second, the 4th to the third, and then on:

1:1 = i.0000, which is less than phi by 0.6180

2:1 = 2.0000, which is 0.3820 more than phi

3:2 = 1.5000, which is less than phi by 0.1180

5:3 = 1.6667, which is 0.0486 more than phi

8:five = one.6000, which is less than phi past 0.0180

As you motion along the Fibonacci summation sequence, each new term volition separate the next with more and more approximation to the unattainable F.

A person subconsciously seeks the Divine proportion: it is needed to satisfy his need for condolement.

When dividing any member of the Fibonacci sequence past the next one, we become merely the reciprocal of i.618 (1: i.618=0.618). But this is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio should also accept no end.

When dividing each number by the adjacent one after it, we go the number 0.382

Selecting ratios in this mode, we obtain the main set up of Fibonacci coefficients: 4.235 ,two.618 ,1.618,0.618,0.382,0.236. Nosotros as well mention 0.v. All of them play special function in nature and in particular in technical analysis.

It should be noted here that Fibonacci only reminded mankind of his sequence, since information technology was known back in aboriginal times called the golden ratio.

The golden ratio, as we accept seen, arises in connexion with the regular pentagon, and therefore the Fibonacci numbers play a role in everything that has to do with regular pentagons - convex and star-shaped.

The Fibonacci series could accept remained only a mathematical incident if information technology were 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 serial every bit an arithmetic expression of the golden sectionalization law. Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich using Fibonacci numbers solves Hilbert's tenth problem (on the solution of Diophantine equations). There are elegant methods for solving a number of cybernetic bug (search theory, games, programming) using Fibonacci numbers and the golden department. In the United states, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this surface area is the discovery of generalized Fibonacci numbers and generalized aureate ratios. The Fibonacci series (ane, 1, ii, 3, five, 8) and the "binary" series of numbers discovered by him ane, 2, 4, 8, 16 ... (that is, a series of numbers upwardly to n, where any natural number, less than n tin be represented by the sum of some numbers of this serial) at first glance, they 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 = one + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers two \u003d 1 + 1, three \u003d two + 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's set a numerical parameter S, which can take any values: 0, 1, ii, 3, four, five... separated from the previous i by S steps. If nth member we denote this series by S (north), then we obtain general formula S (n) \u003d S (due north - ane) + S (n - Southward - ane).

Obviously, with Due south = 0, from this formula we will become a "binary" series, with S = 1 - a Fibonacci series, with S = 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

IN general view the gold S-proportion is the positive root of the golden Southward-section x Due south+1 – x S – 1 = 0.

Information technology is easy to show that at Southward = 0, the division of the segment in half is obtained, and at S = ane, the familiar classical gold section is obtained.

The ratios of neighboring Fibonacci Southward-numbers with absolute mathematical accuracy coincide in the limit with the golden Southward-proportions! That is, golden S-sections are numerical invariants of Fibonacci S-numbers.

vii. Aureate section in art.

7.1. Golden department in painting.

Turning to examples of the "golden section" in painting, one cannot only stop one's attention on the work of Leonardo da Vinci. His identity is one of the mysteries of history. Leonardo da Vinci himself said: "Let no one who is not a mathematician dare to read my works."

There is no doubt that Leonardo da Vinci was a great artist, his contemporaries already recognized this, only his personality and activities will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, just just numerous handwritten sketches, notes that say "both everyone in the world."

The portrait of Monna Lisa (Gioconda) has been alluring the attention of researchers for many years, who plant that the composition of the cartoon is based on golden triangles that are parts of a regular star pentagon.

Likewise, the proportion of the gilt section appears in Shishkin's painting. In this famous painting by I. I. Shishkin, the motifs of the aureate department are clearly visible. The brightly lit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pino tree is a hillock illuminated by the dominicus. It divides the correct side of the picture horizontally according to the golden ratio.

Raphael's painting "The Massacre of the Innocents" shows another element of the gilded ratio - the golden screw. On the preparatory sketch of Raphael, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the kid's ankle - along the figures of the kid, the adult female clutching him to herself, the warrior with a raised sword and then along the figures of the same group on the correct side of the sketch . It is non known whether Raphael built the golden spiral or felt it.

T. Cook used the golden section when analyzing the painting by Sandro Botticelli "The Nativity of Venus".

vii.two. Pyramids of the gilt department.

The medical properties of the pyramids, peculiarly the golden section, are widely known. According to some of the most mutual opinions, the room in which such a pyramid is located seems larger, and the air is more than transparent. Dreams begin to be remembered meliorate. Information technology is as well known that the golden ratio was widely used in architecture and sculpture. An example of this was: the Pantheon and Parthenon in Hellenic republic, the buildings of architects Bazhenov and Malevich

eight. Conclusion.

Information technology must be said that the golden ratio has a great application in our lives.

It has been proven that the man torso is divided in proportion to the aureate ratio by the chugalug line.

The trounce of the nautilus is twisted like a aureate screw.

Thanks to the gilded ratio, the asteroid belt between Mars and Jupiter was discovered - in proportion there should be another planet there.

The excitation of the cord at the bespeak dividing it in relation to the gilt partitioning will not crusade the cord to vibrate, that is, this is the point of compensation.

On shipping with electromagnetic energy sources, rectangular cells with the proportion of the gilt section are created.

Gioconda is congenital on golden triangles, the golden screw is present in Raphael's painting "Massacre of the Innocents".

Proportion found in the painting by Sandro Botticelli "The Nascence of Venus"

There are many architectural monuments congenital 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, owns the statement: "Geometry has 2 great treasures. The get-go is the Pythagorean theorem, the second is the division of a segment in the farthermost and boilerplate ratio"

Bibliography

1. D. Pidow. Geometry and art. – M.: Mir, 1979.

2. Journal "Science and engineering"

3. Magazine "Quantum", 1973, No. eight.

four. Periodical "Mathematics at School", 1994, No. 2; No. iii.

five. Kovalev F.V. Golden section in painting. K .: Vyscha school, 1989.

six. Stakhov A. Codes of the gold ratio.

7. Vorobyov North.N. "Fibonacci numbers" - M.: Nauka 1964

eight. "Mathematics - Encyclopedia for children" Chiliad .: Avanta +, 1998

ix. Information from the Internet.

Fibonacci matrices and the and so-called "gilded" matrices, new figurer arithmetic, a new coding theory and a new theory of cryptography. essence new science, in the revision from the betoken of view of the golden section of all mathematics, starting with Pythagoras, which, of course, will entail in the theory new and probably very interesting mathematical results. In applied terms - "aureate" computerization. And because...

This result volition not be afflicted. The basis of the golden ratio is an invariant of the recursive ratios four and half-dozen. This shows the "stability" of the golden section, one of the principles of the organization of living thing. Likewise, the ground of the aureate ratio is the solution of two exotic recursive sequences (Fig. iv.) Fig. 4 Recursive Fibonacci Sequences So...

The ear is j5 and the distance from ear to crown is j6. Thus, in this statue we encounter a geometric progression with the denominator j: one, j, j2, j3, j4, j5, j6. (Fig. ix). Thus, the aureate ratio is one of the fundamental principles in the art of ancient Greece. Rhythms of the heart and brain. The man heart beats evenly - about threescore beats per minute at residue. The heart compresses like a piston...

June eight, 2011

First way- on this side Due south with the help of a protractor.

Draw a straight line and plot AB = S on it; nosotros accept this line equally a radius and with this radius from points A and B we depict arcs: so, using a protractor, we build angles of 108 ° at these points, the sides of which will intersect with arcs at points C and D; from these points with radius AB = 5 we describe the arcs that intersect at E, and connect the points L, C, E, D, B with direct lines.

The resulting pentagon- desired.

The 2d way. Draw a circle with radius r. From betoken A we draw an arc of radius AM with a compass until it intersects at points B and C with a circle. We connect B and C with a line that will cross the horizontal axis at point E.

Then, from point Eastward, nosotros depict an arc that will intersect the horizontal line at point O. Finally, from indicate F, we describe an arc that volition intersect the circle at points H and K. Having set aside the distance FO \u003d FH \u003d FK v times forth the circle and connecting the division points with lines, we get a regular pentagon.

The 3rd way. Inscribe a regular pentagon in this circle. We describe two mutually perpendicular diameters AB and MC. Divide the radius AO by the point E in half. From point E, as from the centre, we depict an arc of a circle of radius EM and mark the diameter AB at point F with it. The segment MF is equal to the side of the desired regular pentagon. With a compass solution equal to MF, nosotros brand serifs Due north 1, P one, Q 1, Grand 1 and connect them with straight lines.

The effigy shows a hexagon along this side.

Direct AB \u003d 5, as a radius, from points A and B we depict arcs that intersect at C; from this point, with the same radius, we depict a circle on which side A B will exist deposited 6 times.

Hexagon ADEFGB- desired.

"Refurbishment of rooms during renovation",
N.P.Krasnov

The first manner to build. We draw the horizontal (AB) and vertical (CD) axes and from the point of their intersection M nosotros set aside the semi-axes in the advisable scale. We plot the pocket-size semiaxis from point M on the major axis to betoken E. Ellipse, the first construction method Divide BE into 2 parts and apply one from point M on the major axis (to F or H) ...

The basis for applying the painting is the completely finished painting of the surfaces of walls, ceilings and other structures; the painting is done on loftier-quality glue and oil paints, made for trimming or fluting. Starting to develop a sketch of the finish, the master must conspicuously imagine the whole composition in a domestic surround and conspicuously realize the artistic idea. But if this basic condition is observed can one correctly ...

The measurement of the piece of work performed, except in special cases, is carried out according to the area of ​​the actually processed surface, taking into account its relief and minus the untreated places. To decide the really processed surfaces during painting piece of work, y'all should use the conversion factors given in the tables. A. Wooden window devices (measured by the area of ​​​​openings along the outer contour of the boxes) Device name Coefficient for ...

If there is no compass at hand, and so you can draw a unproblematic star with five rays, then simply connect these rays. every bit yous can meet in the film beneath, an absolutely regular pentagon is obtained.

Mathematics is a complex scientific discipline and it has many secrets, some of them are very funny. If yous are interested in such things, I advise you to find the book Funny Math.

A circle tin be drawn non only with a compass. You can, for instance, use a pencil and thread. We measure out the desired diameter on the thread. Nosotros tightly clamp one end on a piece of paper, where we will draw a circle. And on the other end of the thread, the pencil is prepare and obsessed. Now it works like with a compass: we stretch the thread and lightly printing the circumvolve around the circumvolve with a pencil.

Inside the circle, draw peasants from the centre: a vertical line and a horizontal line. The intersection signal of the vertical line and the circle will be the vertex of the pentagon (point 1). At present nosotros divide the correct half of the horizontal line in half (signal 2). We measure the altitude from this point to the vertex of the pentagon and puts this segment to the left of point 2 (point 3). Using a thread and a pencil, we draw an arc from signal 1 with a radius to indicate 3 that intersects the kickoff circumvolve on the left and right - the intersection points will be the vertices of the pentagon. Let'south designate their point 4 and 5.

Now from betoken 4 we make an arc that intersects the circumvolve in the lower part, with a radius equal to the length from point i to iv - this will be point half dozen. Similarly, from point 5 - we will denote point 7.

It remains to connect our pentagon with vertices 1, 5, 7, 6, iv.

I know how to build a simple pentagon using a compass: Describe a circle, mark 5 points, connect them. Y'all can build a pentagon with equal sides, for this we yet need a protractor. Nosotros just put the same five points along the protractor. To practise this, mark the angles of 72 degrees. And so we besides connect with segments and get the figure we need.

The green circle can be drawn with an arbitrary radius. We will inscribe a regular pentagon in this circle. Without a compass, it is incommunicable to draw an exact circle, but this is not necessary. The circumvolve and all further constructions can be washed by mitt. Adjacent, through the center of the circumvolve O, you lot need to describe 2 mutually perpendicular lines and designate one of the points of intersection of the line with the circle A. Point A volition be the vertex of the pentagon. We divide the radius OB in half and put a point C. From signal C we draw a second circle with a radius Air conditioning. From indicate A nosotros draw a third circumvolve with radius AD. The intersection points of the third circle with the get-go (East and F) will likewise be the vertices of the pentagon. From points E and F with radius AE nosotros make notches on the first circumvolve and become the remaining vertices of the pentagon Grand and H.



Adepts of black fine art: in order to but, beautifully and rapidly draw a pentagon, yous should draw a correct, harmonious basis for the pentagram (five-pointed star) and connect the ends of the rays of this star through direct, even lines. If everything was done correctly, the connecting line around the base of operations volition be the desired pentagon.

(in the effigy - a completed merely unfilled pentagram)



For those who are unsure of the right design of the pentagram: take Da Vinci'south Vitruvian Man equally a basis (run into below)



If you need a pentagon, randomly poke the 5th point and their outer contour volition be a pentagon.

If you need a regular pentagon, and then without a mathematical compass this construction is incommunicable, since without it you cannot draw 2 identical, but not parallel, segments. Any other tool that allows you to draw two identical, but not parallel segments is equivalent to a mathematical compass.



Beginning you need to draw a circle, then guides, then the 2nd dotted circumvolve, find the top point, then mensurate the top two corners, draw the bottom ones from them. Notation that the radius of the compass is the same throughout the structure.

It all depends on what kind of pentagon you demand. If any, then put five points and connect them together (naturally, we do non prepare the points in a straight line). And if you need a correctly shaped pentagon, take whatever 5 in length (strips of newspaper, matches, pencils, etc.), lay out the pentagon and outline it.

A pentagon can be fatigued, for example, from a star. If y'all know how to draw a star, just do not know how to draw a pentagon, draw a star with a pencil, so connect the side by side ends of the star together, and so erase the star itself.

The 2d style. Cut out a strip of newspaper with a length equal to the desired side of the pentagon, and a narrow width, say 0.5 - one cm. As per the template, cutting four more than of the aforementioned strips forth this strip to make merely 5 of them.

So put a sheet of paper (it is better to ready information technology on the tabular array with 4 buttons or needles). Then lay these 5 strips on the leaf so that they form a pentagon. Pin these 5 strips to a slice of paper with pins or needles so that they remain motionless. Then circle the resulting pentagon and remove these stripes from the canvass.



If there is no compass and y'all need to build a pentagon, and so I can advise the following. I congenital it myself. Tin can you draw the correct 5-pointed star. And later that, to get a pentagon, you just need to connect all the vertices of the star. This is how the pentagon will plough out. Here's what we'll get



We connected the vertices of the star with fifty-fifty black lines and got a pentagon.

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Source: https://goaravetisyan.ru/en/postroenie-5-ugolnika-v-okruzhnosti-kak-postroit-pyatiugolnik-s/

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