Pentagon with compass and ruler. How to build and draw a regular pentagon around a circle

This shape is a polygon with a minimum number of corners that cannot be paved with an area. Only a 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, inscribe it into a circle with a given radius, or build it on the basis of a given lateral side.

How to draw a ray 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 drag 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 caption, click the "Draw caption" button and create a field where the caption 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 containing 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.

The task of constructing a true pentagon is reduced to the task of dividing a circle into five equal parts. Since the correct pentagon is one of the figures containing the proportions of the golden section, painters and mathematicians have long been interested in its construction. Several methods have now been discovered for constructing a correct polygon inscribed in a given circle.

You will need

• - ruler
• - compass

Instructions

1. Apparently, if you build a correct decagon, and then combine its vertices through one, you get a pentagon. To draw a decagon, draw a circle with a given radius. Mark its center with the letter O. Draw two perpendicular radii, in the figure they are denoted as OA1 and OB. Divide the radius OB in half with the aid of a ruler or by dividing the segment in half with the aid of a compass. Draw a small circle with center C in the middle of OB with a radius equal to half of OB. Join point C with point A1 on the start circle with a ruler. Line segment CA1 intersects the construction circle at point D. Line segment DA1 is equal to the side of the true decagon inscribed in this circle. With a compass, sweep this segment on the circle, then combine the intersection points through one and you will get a positive pentagon.

2. Another method was discovered by the German artist Albrecht Durer. To build a pentagon according to his method, start again by drawing a circle. Re-sweep its center O and draw two perpendicular radii OA and OB. Divide the radius OA in half and sweep the middle with the letter C. Place the compass needle at point C and open it to point B. Draw a circle with radius BC until it intersects the diameter of the starting circle on which the radius OA lies. Designate the intersection point D. Line segment BD is the side of the positive pentagon. Depart this segment five times on the starting circle and combine the intersection points.

3. If you need to build a pentagon along its given side, then you need the 3rd method. Draw the side of the pentagon along the ruler, mark this segment with the letters A and B. Divide it into 6 equal parts. Draw a ray from the middle of segment AB, perpendicular to the segment. Construct two circles with radius AB and centers at A and B, as if you were going to cut the segment in half. These circles intersect at point C. Point C in this case lies on a ray outgoing perpendicularly upward from the middle of AB. Defer from C upward along this ray a distance equal to 4/6 of the length AB, designate this point D. Construct a circle of radius AB centered at point D. The intersection of this circle with the two auxiliary ones constructed earlier will give the last two vertices of the pentagon.

The topic of dividing a circle into equal parts in order to construct correct inscribed polygons has long occupied the minds of ancient scientists. These theses of construction using a compass and a ruler were expressed in the Euclidean "Principles". However, only two millennia later this problem was completely solved not only graphically, but also mathematically.

Instructions

1. Approximate construction of a positive pentagon by the method of A. Dürer, with the help of a compass and a ruler (through two circles with a total radius equal to the side pentagon).

2. Building the Faithful pentagon based on a positive decagon inscribed in a circle (combining the vertices of the decagon through one).

3. Plotting through the calculated internal angle pentagon with the support of a protractor and a ruler (the sum of the angles of a convex n-gon is Sn = 180 ° (n - 2), since all angles of a positive polygon are equal). With n = 5, S5 = 5400, then the value of the angle is 1080; (36005 = 720). Their intersection with a circle will give a segment equal to the side pentagon .

4. Another easy graphical method: divide the diameter of a given circle AB into three parts (AC = CD = DE). From point D lower the perpendicular to the intersection with the circle at points E, F. Drawing straight lines through the segments EC and FC until the intersection with the circle, we obtain points G, H. Points G, E, B, F, H are the vertices of the positive pentagon .

5. Construction with support for Bion's technique (which allows us to construct a true polygon inscribed in a circle with any number of sides n according to a given ratio). Let's say: for n = 5. Let us erect a positive triangle ABC, where AB is the diameter of the given circle. Find point D on AB, according to the further ratio: AD: AB = 2: n. For n = 5, AD = 25 * AB. Let us draw a straight line through CD up to the intersection with the circle at point E. Segment AE is the side of the correct inscribed pentagon. For n = 5,7,9,10, the construction error does not exceed 1%. With increasing n, the approximation error grows, but remains less than 10.3%.

6. Construction on a given side by the method of L. Da Vinci (using the ratio between the side of the polygon (аn) and apothem (ha): аn / 2: ha = 3 / (n-1), which can be expressed as follows: tg180 ° / n = 3 / (n-1)).

7. A general method for constructing positive polygons on a given side according to the method of F. Kovarzhik (1888), based on the rule of L. da Vinci. An integral method for constructing a positive n-gon based on Thales's theorem. It may only be added that approximate methods for constructing polygons are genuine, primitive and beautiful.

There are two main methods for constructing a true five-sided polygon. Both of them suggest the use of a compass, ruler and pencil. 1st method is fitting pentagon in a circle, and the 2nd method is based on a given length of the side of your upcoming geometric figure.

You will need

• Compass, ruler, pencil

Instructions

1. 1st construction method pentagon considered more "typical". First, draw a circle and somehow mark its center (usually the letter O is used for this). After that, draw the diameter of this circle (let's call it AB) and divide one of the 2 obtained radii (say, OA) exactly in half. The middle of this radius will be denoted by the letter C.

2. From point O (center of the initial circle) draw another radius (OD), one that will be strictly perpendicular to the previously drawn diameter (AB). After that, take a compass, put it at point C and measure the distance to the intersection of the new radius with the circle (CD). Set the same distance on the AB diameter. You will get a new point (let's call it E). Measure the distance from point D to point E with a compass - it will be equal to the length of the side of your future pentagon .

3. Place a compass at point D and mark the distance equal to the segment DE on the circle. Repeat this procedure 3 more times, and then combine point D and 4 new points on the starting circle. The resulting shape will be a true pentagon.

4. In order to build a pentagon using a different method, first draw a line segment. Let's say it will be a 9 cm segment AB. Then divide your segment into 6 equal parts. In our case, the length of any part will be 1.5 cm. Now take a compass, place it at one of the ends of the segment and draw a circle or arc with a radius equal to the length of the segment (AB). After that, move the compass to the other end and repeat the operation. The resulting circles (or arcs) will intersect at one point. Let's call it C.

5. Now take a ruler and draw a straight line through point C and the center of line segment AB. After that, starting from point C, set aside on this straight line a segment that is 4/6 of segment AB. The second end of the segment will be denoted by the letter D. Point D will be one of the vertices of the upcoming pentagon... From this point, draw a circle or arc with a radius equal to AB. This circle (arc) will intersect the previously constructed circles (arcs) at points that are two missing vertices pentagon... Combine these points with vertices D, A and B, and plotting a positive pentagon will be finished.

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Ray - it is a straight line drawn from a point and has no end. There are other definitions of a ray: say, "... it is a straight line bounded by a point on one side." How to positively trace a ray, and what drawing supplies do you need?

You will need

• A sheet of paper, a pencil and a ruler.

Instructions

1. Take a piece of paper and sweep a dot anywhere. After that, attach a ruler and draw a line from the indicated point to infinity. This drawn line is called a ray. Now mark another point on the ray, for example, the letter C. The line from the origin to point C will be called a segment. If you draw a line primitively and do not really sweep one point, then this line will not be a ray.

2. It is no more difficult to draw a ray in any graphics editor or in the same MSOffice than manually. Take Microsoft Office 2010 as an example. Go to Insert and select Shapes. Select the Line shape from the drop-down list. Further, the cursor will take the form of a cross. To draw a straight line, press the Shift key and draw a line of the required length. The Format tab will open immediately after the style. Now you have drawn a primitive straight line and there is no fixed point, and based on the definition, the ray should be limited to a point on one side.

3. To make a point at the beginning of a line, do the following: select the drawn line and call the context menu by pressing the right mouse button.

4. Select Shape Format. In the menu on the left, select "Line Type". Next, find the heading "Line Parameters" and select the "Start Type" in the form of a circle. There you can also adjust the thickness of the start and end lines.

5. Remove the selection from the line and you will see that a point appears at the beginning of the line. To create an inscription, press the "Draw inscription" button and make a field where the inscription will be located. After writing the inscription, click on the free space and it will be activated.

6. The beam is safely drawn and it took every few minutes. Drawing a ray in other editors is carried out according to the same thesis. While holding down the Shift key, proportional shapes will be drawn invariably. Glorious use.

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Note!
The ratio of the diagonal of a true pentagon to its side is the golden ratio (irrational number (1 + √5) / 2). All of the five interior angles of the pentagon are 108 °.

Helpful advice
If you combine the vertices of a true pentagon with diagonals, you get a pentagram.

A regular pentagon can be constructed using a compass and a ruler, or by inscribing it into a given circle, or by building on a given side. This process is described by Euclid in his "Beginnings" about 300 BC. NS.

Here is one method for constructing a regular pentagon in a given circle:

1. Construct a circle in which the pentagon will be inscribed and mark its center as O... (This is the green circle in the diagram on the right).

1. Select a point on the circle A, which will be one of the vertices of the pentagon. Build a straight line through O and A.
2. Construct a line perpendicular to a line OA passing through the point O... Designate one of its intersections with a circle as a point B.
3. Build a point C in the middle between O and B.
4. C through the point A... Mark its intersection with a straight line OB(inside the original circle) as a point D.
5. Draw a circle centered at A through the point D... Mark its intersections with the original (green circle) as points E and F.
6. Draw a circle centered at E through the point A G.
7. Draw a circle centered at F through the point A... Designate its other intersection with the original circle as a point H.
8. Construct a regular pentagon AEGHF.

Icosahedron

Icosahedron- one of the five Platonic solids, in simplicity following the tetrahedron and octahedron. They are united by the fact that the faces of each are equilateral triangles. When making an icosahedron model, you can choose between two spectacular distribution options for the five colors.

First, the icosahedron can be colored so that each vertex has all five colors (however, in this case, the opposite faces will not be colored the same).

Another method provides the opposite colors with the same colors, but at each vertex, except for two polar ones, the same color will be repeated in a circle. Both colorings are very interesting and useful for our purposes, because many of the homogeneous polyhedra described below have icosahedral symmetry.

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 draw 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, how to draw a star using a ruler, you must have knowledge of the simplest mathematical formulas that are fundamental in geometry. You will also need to be able 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 figure show through.

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, you should carry out all actions 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 a 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 complex mathematical calculations, it is possible to obtain 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. Using fantasy and imagination, you can create an original composition from the obtained correct, beautiful form. 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!

Difficulty level: Easy

Step 1

First, choose where to place the center of the circle. There you need to put a starting point, let it be called O. With the help of a compass, draw a circle of a given diameter or radius around it.

Step 2

Then we draw two axes through point O, the center of the circle, one horizontal, the other at 90 degrees in relation to it - vertical. The intersection points horizontally will be called from left to right A and B, vertically, from top to bottom - M and N. The radius that lies on any axis, for example, on the horizontal on the right, is halved. This can be done as follows: a compass with the radius of a circle known to us is set with the tip at the intersection of the horizontal axis and the circle - B, we delineate the intersections with the circle, we call the resulting points, respectively, from top to bottom - C and P, connect them with a segment that will intersect the OB axis, the intersection point is called K.

Step 3

We connect points K and M and get a segment KM, set a compass at point M, set the distance to point K on it and outline marks on the radius OA, call this point E, then draw a compass until it intersects with the upper left part of the circle OM. This intersection point is called F. The distance equal to the segment ME is the desired side of the equilateral pentagon. In this case, point M will be one vertex of the pentagon embedded in the circle, and point F will be another.

Step 4

Further, from the obtained points along the entire circle, we draw with a compass the distances equal to the ME segment, the total points should be 5. We connect all the points with segments - we get a pentagon inscribed in the circle.

• When drawing, be careful in measuring distances, do not allow errors, so that the pentagon really turns out to be equilateral