If there is no compass at hand, then you can draw a simple star with five rays, then simply connect these rays. as you can see in the picture below, an absolutely regular pentagon is obtained.
Mathematics is a complex science and it has many secrets, some of them are very funny. If you are interested in 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 desired diameter on the thread. We 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 set and obsessed. Now it works like with a compass: we stretch the thread and lightly press the circle around the circle with a pencil.
Inside the circle, draw peasants from the center: a vertical line and a horizontal line. The intersection point of the vertical line and the circle will be the vertex of the pentagon (point 1). Now we 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 puts this segment to the left of point 2 (point 3). Using a thread and a pencil, we draw an arc from point 1 with a radius to point 3 that intersects the first circle on the left and right - the intersection points will be the vertices of the pentagon. Let's designate their point 4 and 5.
Now from point 4 we make an arc that intersects the circle in the lower part, with a radius equal to the length from point 1 to 4 - this will be point 6. Similarly, from point 5 - we will denote 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. Then we also 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 impossible to draw an exact circle, but this is not necessary. The circle and all further constructions can be done by hand. Next, through the center of the circle O, you need to draw two mutually perpendicular lines and designate one of the points of intersection of the line with the circle A. Point A will be the vertex of the pentagon. We divide the radius OB in half and put a point C. From point C we draw a second circle with a radius AC. From point A we draw a third circle with radius AD. The intersection points 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 notches on the first circle and get the remaining vertices of the pentagon G and H.
Adepts of black art: in order to simply, beautifully and quickly draw a pentagon, you should draw a correct, harmonious basis for the pentagram (five-pointed star) and connect the ends of the rays of this star with straight, even lines. If everything was done correctly, the connecting line around the base will be the desired pentagon.
(in the figure - a completed but unfilled pentagram)
For those who are unsure of the correct design of the pentagram: take Da Vinci's Vitruvian Man as a basis (see below)
If you need a pentagon, randomly poke the 5th point 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 you cannot draw two 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.
First you need to draw a circle, then guides, then the second dotted circle, find the top point, then measure the top two corners, draw the bottom ones from them. Note that the radius of the compass is the same throughout the construction.
It all depends on what kind of pentagon you need. If any, then put five points and connect them together (naturally, we do not set the points in a straight line). And if you need a correctly shaped pentagon, 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 do not 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.
The second way. Cut out 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 per the template, cut four more of the same strips along this strip to make only 5 of them.
Then put a sheet of paper (it is better to fix it on the table with four buttons or needles). Then lay these 5 strips on the leaf so that they form a pentagon. Pin these 5 strips to a piece of paper with pins or needles so that they remain motionless. 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 the correct 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'll get
We connected the vertices of the star with even black lines and got a pentagon.
A regular pentagon is a geometric figure that is formed by the intersection of five straight lines that create five identical angles. This figure is called the Pentagon. The work of artists is closely related to the pentagon - their drawings are built on the basis of correct geometric shapes. To do this, you need to know how to quickly build a pentagon.
Why is this figure interesting? The building is shaped like a pentagon Department of Defense of the United States of America. This can be seen in the photos taken from the height of the flight. In nature, there are no crystals and stones, the shape of which would resemble a pentagon. Only in this figure the number of faces coincides with the number of diagonals.
Parameters of a regular pentagon
A rectangular pentagon, like every figure in geometry, has its own parameters. Knowing the necessary formulas, you can calculate these parameters, which will facilitate the process of building a pentagon. Calculation methods and formulas:
- the sum of all angles in polygons is 360 degrees. In a regular pentagon, all angles are equal, respectively, central corner is found in this way: 360/5 \u003d 72 degrees;
- the inner corner is found in this way: 180*(n -2)/ n = 180*(5−2)/5 = 108 degrees. The sum of all interior angles: 108*5 = 540 degrees.
The side of the pentagon is found using the parameters that are already given in the problem statement:
- if a circle is circumscribed around the pentagon and its radius is known, the side is found according to the following formula: a \u003d 2 * R * sin (α / 2) \u003d 2 * R * sin (72/2) \u003d 1.1756 * R.
- If the radius of the circle inscribed in the pentagon is known, then the formula for calculating the side of the polygon is: 2*r*tg (α/2) = 2*r*tg (α/2) = 1.453*r.
- With a known diagonal of the pentagon, its side is calculated as follows: a \u003d D / 1.618.
The area of the pentagon, like its side, depends on the parameters already found:
- using the known radius of the inscribed circle, the area is found as follows: S \u003d (n * a * r) / 2 \u003d 2.5 * a * r.
- the circumscribed circle around the pentagon allows you to find the area using the following formula: S \u003d (n * R2 * sin α) / 2 \u003d 2.3776 * R2.
- depending on the side of the pentagon: S = (5*a2*tg 54°)/4 = 1.7205* a2.
Building the Pentagon
You can build a regular pentagon using a ruler and a compass, based on a circle inscribed in it or one of the sides.
How to draw a pentagon based on an inscribed circle? To do this, stock up on a compass and a ruler and take the following steps:
- First you need to draw a circle with center O, then select a point on it, A - the top of the pentagon. A line is drawn from the center to the top.
- Then a segment perpendicular to the straight line OA is constructed, which also passes through O - the center of the circle. Its intersection with the circle is indicated by point B. The segment O.V. is bisected by point C.
- Point C will become the center of a new circle passing through A. Point D is its intersection with the straight line OB within the boundaries of the first figure.
- After that, a third circle is drawn through D, the center of which is point A. It intersects with the first figure at two points, they must be denoted by the letters E and F.
- The next circle has its center at point E and passes through A, and its intersection with the original one is at the new point G.
- The last circle in this figure is drawn through a point, A with a center F. Point H is placed at its intersection with the initial one.
- On the first circle, after all the steps taken, five points appeared, which must be connected by segments. Thus, a regular pentagon AE G H F was obtained.
How to build a regular pentagon in a different way? With the help of a ruler and a compass, the pentagon can be built a little faster. For this you need:
- First you need to use a compass to draw a circle, the center of which is point O.
- The radius OA is drawn - a segment that is plotted on a circle. It is bisected by point B.
- A segment OS is drawn perpendicular to the radius OA, points B and C are connected by a straight line.
- The next step is to plot the length of segment BC with a compass on the diametral line. Point D appears perpendicular to segment OA. Points B and D are connected, forming a new segment.
- In order to get the size of the side of the pentagon, you need to connect points C and D.
- D with the help of a compass is transferred to the circle and is indicated by the point E. By connecting E and C, you can get the first side regular pentagon. Following this instruction, you can learn how to quickly build a pentagon with equal sides, continuing to build its other sides like the first one.
In a pentagon with the same sides, the diagonals are equal and form a five-pointed star, which is called a pentagram. golden ratio is the ratio of the diagonal to the side of the pentagon.
The Pentagon is not suitable for completely filling the plane. The use of any material in this form leaves gaps or forms overlaps. Although natural crystals of this form do not exist in nature, when ice forms on the surface of smooth copper products, molecules in the form of a pentagon appear, which are connected in chains.
The easiest way to get a regular pentagon from a strip of paper is to tie it in a knot and press down a little. This method is useful for parents of preschoolers who want to teach their toddlers to recognize geometric shapes.
Video
See how you can quickly draw a pentagon.
Construction of a regular hexagon inscribed in a circle.
The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to construct, it is enough to divide the circle into six equal parts and connect the found points to each other.
A regular hexagon can be constructed using a T-square and a 30X60° square. To perform this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4, build sides 1 - 6, 4 - 3, 4 - 5 and 7 - 2, after which we draw sides 5 - 6 and 3 - 2.
The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60 °, or only one compass. Consider two ways to construct an equilateral triangle inscribed in a circle.
First way(Fig. 61, a) is based on the fact that all three angles of the triangle 7, 2, 3 each contain 60 °, and the vertical line drawn through the point 7 is both the height and the bisector of angle 1. Since the angle 0 - 1 - 2 is equal to 30°, then to find the side 1 - 2 it is enough to construct an angle of 30° from point 1 and side 0 - 1. To do this, set the T-square and square as shown in the figure, draw a line 1 - 2, which will be one 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.
Second way is based on the fact that if you build a regular hexagon inscribed in a circle, and then connect its vertices through one, you get an equilateral triangle.
To build a triangle, we mark the vertex point 1 on the diameter and draw a diametrical line 1 - 4. Further, from point 4 with a radius equal to D / 2, we describe the arc until it intersects with the circle at points 3 and 2. The resulting points will be two other vertices of the desired triangle.
This construction can be done using a square and a compass.
First way is based on the fact that the diagonals of the square intersect at the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install a T-square and a square 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 square 4 - 1 and 3 -2 with the help of a T-square. Then, using a T-square along the leg of the square, we draw the vertical sides of the square 1 - 2 and 4 - 3.
Second way is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter. 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 draw auxiliary lines, marked on the figure with solid lines. Their points of intersection with the circle will define vertices 1 and 3; 4 and 2. The vertices of the desired square obtained in this way are connected in series with each other.
Construction of a regular pentagon inscribed in a circle.
To inscribe a regular pentagon in a circle, we make the following constructions. We mark point 1 on the circle and take it as one of the vertices of the pentagon. Divide segment AO in half. To do this, with the radius AO from point A, we describe the arc to the intersection with the circle at points M and B. Connecting these points with a straight line, we get the point K, which we then connect to point 1. With a radius equal to the segment A7, we describe the arc from point K to the intersection with the diametrical line AO at point H. Connecting point 1 with point H, we get the side of the pentagon. Then, with a compass opening equal to the segment 1H, describing the arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass opening, we obtain the remaining vertices 3 and 4. 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 six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on the line AB we draw a vertical line. Further from the point K on this straight line, we set aside a segment equal to 4/6 AB. We get point 1 - the vertex of the pentagon. Then, with a radius equal to AB, from point 1 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 2 and 5. We connect the found vertices in series with each other.
Construction of a regular heptagon inscribed in a circle.
Let a circle of diameter D be given; you need to inscribe a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of the circle D, we describe the arc until it intersects with the continuation of the horizontal diameter at point F. Point F is called the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / - // - /// from points IV, V and VI, we draw horizontal lines until they intersect with the circle. We connect the found vertices in series with each other. The heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.
The above method is suitable for constructing regular polygons with any number of sides.
The division of a circle into any number of equal parts can also be done using the data in Table. 2, which shows the coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.
Side lengths of regular inscribed polygons.
The first column of this table shows the number of sides of a regular inscribed polygon, and the second column shows the coefficients. The length of a side of a given polygon is obtained by multiplying the radius of a given circle by a factor corresponding to the number of sides of this polygon.
Difficulty level: Easy
1 step
First, choose where to place the center of the circle. There you need to put a starting point, let it be called O. Using a compass, draw a circle around it with a given diameter or radius.
2 step
Then we draw two axes through the point O, the center of the circle, one horizontal, the other at 90 degrees in relation to it - vertical. We will call the points of intersection horizontally from left to right A and B, vertically, from top to bottom - M and H. The radius, which lies on any axis, for example, on the horizontal on the right side, is divided in half. This can be done as follows: we set a compass with the radius of the circle known to us with a tip at the intersection point of the horizontal axis and the circle - B, we mark the intersections with the circle, we call the resulting points, respectively, from top to bottom - C and P, we connect them with a segment that will intersect the axis OB, the point of intersection is called K.
3 step
We connect the points K and M and get the KM segment, set the compass to the point M, set the distance to the point K on it and draw marks on the radius OA, call this point E, then we draw the compass to the intersection with the upper left part of the circle OM. We call this point of intersection F. The distance equal to the segment ME is the desired side of the equilateral pentagon. In this case, the point M will be one vertex of the pentagon embedded in the circle, and the point F will be the other.
4 step
Further, from the points obtained along the entire circle, we draw with a compass distances equal to the segment ME, in total there should be 5 points. We connect all the points with segments - we get a pentagon inscribed in a circle.
- When drawing, be careful in measuring distances, do not make mistakes so that the pentagon is really equilateral
Positive pentagon is a polygon in which all five sides and all five angles are equal. It is easy to describe a circle around it. Erect pentagon and this circle will help.
Instruction
1. First of all, you need to build a circle with a compass. Let the center of the circle coincide with point O. Draw axes of symmetry perpendicular each friend. At the intersection point of one of these axes with the circle, put a point V. This point will be the top of the future pentagon A. Place point D at the point of intersection of another axis with the circle.
2. On the segment OD, find the middle and mark point A in it. Later, you need to draw a circle with a compass centered at this point. In addition, it must pass through the point V, that is, with radius CV. Designate the point of intersection of the axis of symmetry and this circle as B.
3. Later, with the help compass draw a circle of the same radius, placing the needle at point V. Designate the intersection of this circle with the original one as point F. This point will become the 2nd vertex of the future true pentagon A.
4. Now it is necessary to draw the same circle through point E, but with the center at F. Designate the intersection of the circle just drawn with the original one as point G. This point will also become one of the vertices pentagon A. Similarly, you need to build another circle. Its center is in G. Let it intersect with the original circle H. This is the last vertex of a true polygon.
5. You should have five vertices. It remains easy to combine them along the line. As a result of all these operations, you will get a positive inscribed in a circle. pentagon .
Building positive pentagons allowed with the support of a compass and straightedge. True, the process is rather long, as, however, is the construction of any positive polygon with an odd number of sides. Modern computer programs allowed to do this in a few seconds.
You will need
- - A computer with AutoCAD software.
Instruction
1. Find the top menu in the AutoCAD program, and in it the "Basic" tab. Click on it with the left mouse button. The Draw panel appears. Various types of lines will appear. Select a closed polyline. It is a polygon, it remains only to enter the parameters. AutoCAD. Allows you to draw a variety of regular polygons. The number of sides can be up to 1024. You can also use the command line, depending on the version, by typing "_polygon" or "multi-angle".
2. Regardless of whether you use 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. Enter the coordinates in the box that appears. It is allowed to denote them as (0,0), but there may be any other data.
3. Select the required construction method. . AutoCAD offers three options. The pentagon can be circumscribed around a circle or inscribed in it, but it is allowed to build it along given size sides. Select the desired option and press enter. If necessary, set the radius of the circle and also press enter.
4. A pentagon on a given side is first constructed correctly in the same way. Select Draw, a closed polyline, and enter the number of sides. Right-click to open the context menu. Press the command "edge" or "side". In the command line, type the coordinates of the initial and final points of one of the sides of the pentagon. Later 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 "c". In the English version it will be "_e". In order to build an inscribed or circumscribed pentagon, enter later the number of sides of the letter "o" or "c" (or the English "_s" or "_i")
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Helpful advice
With such a simple method, it is possible to build not only a pentagon. In order to construct a triangle, you need to spread the legs of the compass to a distance equal to the radius of the circle. After that, place the needle at any point. Draw a thin auxiliary circle. Two points of intersection of the circles, as well as the point where the leg of the compass was, form three vertices of a positive triangle.
