How to build an equilateral pentagon using a compass. Regular pentagon: the required minimum of information

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, the resulting points are called, respectively, from top to bottom - C and P, we 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 draw 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

Regular pentagon is a geometric figure that is formed by the intersection of five straight lines that create five equal corners. Such a figure is called the Pentagon. The work of artists is closely related to the pentagon - their drawings are based on correct geometric shapes... To do this, you need to know how to quickly build a pentagon.

What is interesting about this figure? The building is shaped like a pentagon United States Department of Defense... This can be seen in the photos taken from the flight altitude. In nature, there are no crystals and stones, the shape of which would resemble the pentagon. Only in this figure, the number of faces is the same as the number of diagonals.

Regular pentagon options

A rectangular pentagon, like every shape in geometry, has its own parameters. Knowing the necessary formulas, you can calculate these parameters, which will facilitate the process of building the pentagon. Calculation methods and formulas:

• the sum of all angles in polygons is 360 degrees. In a regular pentagon, all angles are equal, respectively, the central angle is found in this way: 360/5 = 72 degrees;
• the internal angle is found in this way: 180 * (n -2) / n = 180 * (5-2) / 5 = 108 degrees. The sum of all internal 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 described around the pentagon and its radius is known, the side is found by the following formula: a = 2 * R * sin (α / 2) = 2 * R * sin (72/2) = 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: 2 * r * tg (α / 2) = 2 * r * tg (α / 2) = 1.453 * r.
• With a known value of the diagonal of the pentagon, its side is calculated as follows: a = D / 1.618.

The area of ​​the pentagon is the same, like its side, depends on the parameters already found:

• using the known radius of the inscribed circle, the area is found as follows: S = (n * a * r) / 2 = 2.5 * a * r.
• the circle described around the pentagon allows you to find the area according to the following formula: S = (n * R2 * sin α) / 2 = 2.3776 * R2.
• depending on the side of the pentagon: S = (5 * a2 * tan 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 do I draw a pentagon based on an inscribed circle? To do this, you need to stock up on a compass and a ruler and take the following steps:

1. First, you need to draw a circle with the center O, then select a point on it, A - the top of the pentagon. A segment is drawn from the center to the top.
2. Then a segment perpendicular to the line OA is constructed, which also passes through O - the center of the circle. Its intersection with the circle is denoted by point B. The segment O. V. is divided in half by point C.
3. Point C will become the center of a new circle passing through A. Point D is its intersection with line OB within the boundaries of the first figure.
4. 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.
5. The next circle has a center at point E and passes through A, and its intersection with the original is at a new point G.
6. The last circle in this figure is drawn through point A with center F. Point H is placed at its intersection with the initial one.
7. On the first circle, after all the steps taken, five points appeared, which must be connected with segments. Thus, we got a regular pentagon AE G H F.

How to build a regular pentagon in a different way? With a ruler and compass, the pentagon can be built a little faster. This requires:

1. First, you need to draw a circle with the help of a compass, the center of which is point O.
2. The radius OA is drawn - a segment that is laid down on a circle. It is halved by point B.
3. A segment OS is drawn perpendicular to the radius OA, points B and C are connected by a straight line.
4. The next step is to plot the length of the BC segment using a compass on the centerline. Point D appears perpendicular to line OA. Points B and D are connected to form a new line.
5. In order to obtain the size of the side of the pentagon, it is necessary to connect points C and D.
6. D with the help of a compass is transferred to a circle and is denoted by point E. By connecting E and C, you can get the first side of a regular pentagon. By following this instruction, you can learn how to quickly build a pentagon with equal sides, continuing to build the rest of its sides like the first.

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 unsuitable for completely filling the plane. The use of any material in this form leaves gaps or overlaps. Although natural crystals of this form do not exist in nature, when ice forms on the surface of smooth copper products, pentagon-shaped molecules appear, which are connected into 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.

If there is no compass at hand, then you can draw a simple star with five rays and then simply connect these rays. as you can see in the picture below, an absolutely regular pentagon is obtained.

Mathematics is a difficult science and it has many secrets of its own, some of which are quite amusing. If you are fond of such things, I advise you to find the book Funny Math.

A circle can be drawn not only with a compass. You can, for example, use a pencil and thread. We measure the required diameter on the thread. We clamp one end tightly on a sheet of paper, where we will draw a circle. And on the other end of the thread set a pencil and obsessed. Now it acts as with a compass: we pull the thread and around the circumference, slightly pressing with a pencil, we draw a circle.

Draw peasants from the center inside the circle: a vertical line and a horizontal line. The intersection of the vertical line and the circle will be the vertex of the pentagon (point 1). Now divide the right half of the horizontal line in half (point 2). We measure the distance from this point to the top of the pentagon and this segment is laid to the left of point 2 (point 3). Using a thread and a pencil, draw an arc from point 1 with a radius to point 3, intersecting the first circle on the left and right - the intersection points will be the vertices of the pentagon. Let's designate their points as 4 and 5.

Now from point 4 we make an arc that intersects the circle at the bottom, with a radius equal to the length from point 1 to 4 - this will be point 6. In the same way, from point 5 - we will denote it by point 7.

It remains to connect our pentagon with vertices 1, 5, 7, 6, 4.

I know how to build a simple pentagon using a compass: Draw a circle, mark five points, connect them. You can build a pentagon with equal sides, for this we still need a protractor. We just put the same 5 points along the protractor. To do this, mark the angles of 72 degrees. Then we also connect with segments and get the shape we need.

The green circle can be drawn with an arbitrary radius. We will inscribe a regular pentagon into this circle. It is impossible to draw a circle exactly without a compass, but it is not necessary. The circle and all further construction can be done by hand. Further, through the center of the circle O, you need to draw two mutually perpendicular lines and denote one of the intersection points of the line with the circle A. Point A will be the vertex of the pentagon. We divide the radius of the OB in half and put the point C. From the point C we draw the second circle with the radius AC. From point A we draw a third circle with radius AD. The points of intersection of the third circle with the first (E and F) will also be the vertices of the pentagon. From points E and F with radius AE we make serifs on the first circle and get the remaining vertices of the pentagon G and H.



Adepts of black art: in order to draw a pentagon simply, beautifully and quickly, you should draw a correct, harmonious basis for the pentagram (five-pointed star) and connect the ends of the rays of this star by means of straight, even lines. If everything was done correctly, the connecting line around the base will be the desired pentagon.

(in the figure - a completed, but unfilled pentagram)



For those who are unsure of the correctness of the pentagram: take as a basis the Vitruvian man of Da Vinci (see below)



If you need a pentagon - poke randomly 5 points and their outer contour will be a pentagon.

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



First you need to draw a circle, then the guides, then the second dotted circle, find the top point, then measure the two upper corners, draw the lower ones from them. Notice that the radius of the compass is the same throughout the construction.

It all depends on which pentagon you need. If any, then put five dots and connect them to each other (naturally, we do not put the dots in a straight line). And if you need a pentagon of the correct shape, take any five in length (strips of paper, matches, pencils, etc.), lay out the pentagon and outline it.

A pentagon can be drawn, for example, from a star. If you know how to draw a star, but don't know how to draw a pentagon, draw a star with a pencil, then connect the adjacent ends of the star together, and then erase the star itself.

Second way. Cut a strip of paper with a length equal to the desired side of the pentagon, and a narrow width, say 0.5 - 1 cm.As a template, cut four more of the same stripes along this strip to make a total of 5.

Then put a sheet of paper (it is better to fix it on the table with four buttons or needles). Then place these 5 stripes on a piece of paper so that they form a pentagon. Pin these 5 stripes to a piece of paper with buttons or needles to keep them stationary. Then circle the resulting pentagon and remove these stripes from the sheet.



If there is no compass and you need to build a pentagon, then I can advise the following. I built it myself. You can draw a regular five-pointed star. And after that, to get a pentagon, you just need to connect all the vertices of the star. This is how the pentagon will turn out. Here's what we get



With straight black lines, we connected the vertices of the star and got a pentagon.

Creates 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, for construction, it is enough to divide the circle into six equal parts and connect the found points to each other (Fig. 60, a).

A regular hexagon can be built using a rail and a 30X60 ° square. To perform this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4 (Fig. 60, b), build sides 1 -6, 4-3, 4-5 and 7-2, after which we draw sides 5-6 and 3- 2.

Construction of an equilateral triangle inscribed in a circle... 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.

Consider two ways of constructing an equilateral triangle inscribed in a circle.

The first way(Fig. 61, a) is based on the fact that all three angles of triangle 7, 2, 3 contain 60 ° each, and the vertical line drawn through 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 build an angle of 30 ° along point 1 and side 0-1. To do this, set the raceway and the 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 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.

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 construct a triangle (Fig. 61, b), mark the vertex-point 1 on the diameter and draw the diametrical line 1-4. Further, from point 4 with a radius equal to D / 2, we describe an arc up to the intersection with a circle at points 3 and 2. The resulting points will be the other two vertices of the desired triangle.

Constructing a square inscribed in a circle... This construction 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 circumscribed circle and are inclined to its axes at an angle of 45 °. Based on this, we install the flight tire and the square with angles of 45 ° as shown in Fig. 62, a, and mark points 1 and 3. Next, through these points, we draw the horizontal sides of the square 4-1 and 3-2 with the help of a flight tire. Then, with the help of the racer along the leg of the square, we draw the vertical sides of the square 1-2 and 4-3.

The second method is based on the fact that the vertices of the square are halved by the arcs of a circle enclosed between the ends of the diameter (Fig. 62, b). We mark at the ends of two mutually perpendicular diameters points A, B and C and from them with a radius y we describe arcs until they intersect.

Next, through the points of intersection of the arcs, draw auxiliary straight lines marked on the figure with solid lines. The points of their intersection with the circle will define vertices 1 and 3; 4 and 2. The thus obtained vertices of the required square are connected in series with each other.

Construction of a regular pentagon inscribed in a circle.

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

We mark point 1 on the circle and take it as one of the vertices of the pentagon. Divide the AO segment in half. To do this, with the radius AO from point A, we describe the arc up to the intersection with the circle at points M and B. Connecting these points with a straight line, we get 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 diametral line AO ​​at point H. Connecting point 1 with point H, we get the side of the pentagon. Then, with a compass solution equal to the segment 1H, describing an arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Making the same compass solution the notches from vertices 2 and 5, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Constructs a regular pentagon along its given 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 by straight line AB we draw a vertical line.

We get point 1-vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe an arc until it intersects the arcs previously drawn from points A and B. The intersection points of the arcs define the vertices of 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). We 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 an arc up to the intersection with the continuation of the horizontal diameter at point F. Point F will be 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 the vertices / - // - /// from points IV, V and VI, draw horizontal lines to the intersection with the circle. 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.

The given 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 made 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.

This shape is a polygon with a minimum number of corners that cannot be paved with an area. Only the pentagon has the same number of diagonals as the number of its sides. Using the formulas for an arbitrary regular polygon, you can determine all the necessary parameters that the pentagon has. For example, 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 draw a line of the desired length. Immediately after drawing, the Format tab opens. Remove the selection from the line and you will see a dot appear at the beginning of the line. To create a 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 failed. 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 exactly, 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). Pin these 5 stripes to a piece of paper with buttons or needles 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.