Build a pentagon in a circle using a compass. Detailed pentagon construction. Receiving with a strip of paper
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 can draw a star, but you don’t know how to draw a pentagon, draw a star with a pencil, then connect the adjacent ends of the star, 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.
Positive pentagon Is a polygon with all five sides and all five angles equal to each other. It is easy to describe a circle around it. Erect pentagon and it is this circle that will help.
Instructions
1. First of all, you need to build a circle with a compass. Let the center of the circle coincide with point O. Draw the axes of symmetry perpendicular to each other. At the point of intersection of one of these axes with a circle, put a point V. This point will be the vertex of the future pentagon a. At the point of intersection of another axis with the circle, place point D.
2. On the segment OD, find the middle and mark point A in it. Later, you need to draw a circle with a compass with a center at this point. In addition, it must pass through the point V, that is, the radius CV. The point of intersection of the axis of symmetry and this circle is designated by B.
3. Later, with the help of compasses draw a circle of the same radius, placing a needle at point V. The intersection of this circle with the original one is designated as point F. This point will become the 2nd vertex of the future faithful pentagon a.
4. Now you need to draw the same circle through point E, but with the center at F. The intersection of the just drawn circle with the original one is designated as point G. This point will also become one of the vertices pentagon a. Similarly, you need to build another circle. Its center is G. Let the point of its intersection with the initial circle be H. This is the last vertex of the correct polygon.
5. You should have five peaks. It remains to be easy to combine them according to the ruler. As a result of all these operations, you will receive a positive inscribed in a circle pentagon .
Building positive pentagons allowed with the support of a compass and a ruler. True, this process is quite long, as, however, is the construction of any positive polygon with an odd number of sides. Modern computer programs allow you to do this in a few seconds.
You will need
- - computer with AutoCAD program.
Instructions
1. Find the top menu in AutoCAD and the Main tab in it. Click on it with the left mouse button. The Draw panel appears. Various line types appear. Select the closed polyline. It is a polygon, you just need to enter the parameters. AutoCAD. Allows you to draw a variety of regular polygons. The number of sides can be up to 1024. It is also allowed to use the command line, depending on the version, by typing "_polygon" or "plural-angle".
2. Regardless of whether you are using the command line or context menus, you will see a window on the screen in which you are prompted to enter the number of sides. Enter the number "5" there and press Enter. You will be prompted to determine the center of the pentagon. Type the coordinates into the window that appears. It is allowed to designate them as (0,0), but there can be any other data.
3. Select the required construction method. ... AutoCAD offers three options. A pentagon can be circumscribed around a circle or inscribed in it, but it is allowed to erect it according to a given side size. Select the option you want and press enter. If necessary, set the radius of the circle and also press enter.
4. A pentagon along a given side is first constructed correctly in the same way. Select Draw, a closed polyline, and enter the number of sides. Call the context menu with the right mouse button. Press the "edge" or "side" command. At the command line, type the coordinates of the starting and ending points of one of the sides of the pentagon. Later on, a pentagon will appear on the screen.
5. All operations can be performed with command line support. For example, to build a pentagon along the side in the Russian version of the program, enter the letter "c". In the English version, it will be "_e". In order to build an inscribed or circumscribed pentagon, enter later the definition of the number of sides of the letter "o" or "in" (or the English "_s" or "_i")
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Useful advice
With such a simple method, it is allowed to build not only a pentagon. In order to build a triangle, you need to spread the legs of the compass to a distance equal to the radius of the circle. After that, install the needle at any point. Draw a thin construction circle. Two points of intersection of the circles, as well as the point at which the leg of the compass was, form three vertices of a positive triangle.
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, we 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
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 a 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 = 360 / n = 360/5 = 72 °.
- Internal angle β = 180 ° * (n-2) / n = 180 ° * 3/5 = 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 = 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 * tan (α / 2) = 2 * r * tan (α / 2) ≈ 1.453 * r;
- it so 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 a 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 = (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 lateral side a:
S = (5 * a 2 * tg54 °) / 4 ≈ 1.7205 * a 2.
Regular pentagon: construction
This geometric shape can be constructed in different ways. For example, 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 straight line through point O perpendicular to line OA. The 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 should 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 centered at F. Mark the other intersection of the original circle with H.
9. Now it remains only to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!
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!
