Creation of regular polygons. Constructing regular polygons - technical drawing How to build an 8 gon using a compass

Kuklin Alexey

The work is abstract in nature with elements research activities... It discusses various ways to construct regular n-gons. The paper contains a detailed answer to the question of whether it is always possible to construct an n-gon using a compass and a ruler. The work is accompanied by a presentation, which can be found on this mini-site.

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Construction of regular polygons The work was completed by: a student of grade 9 "B" MBOU secondary school number 10 Kuklin Alexey

Regular polygons A regular polygon is a convex polygon in which all sides and angles are equal. Go to examples A convex polygon is a polygon, all points of which lie on one side of any straight line passing through two of its adjacent vertices.

Back Regular polygons

Ancient Greek scientists were the founders of the section of mathematics on regular polygons. Some of them were Archimedes and Euclid.

Proof of the existence of a regular n-gon If n (the number of corners of the polygon) is greater than 2, then such a polygon exists. Let's try to build an 8 gon and prove it. Proof

Take a circle of arbitrary radius centered at point O. Divide it into a number of equal arcs, in our case 8. To do this, draw the radii so that we get 8 arcs, and the angle between the two nearest radii was 360 °: the number of sides (in our case 8), respectively, each angle will be 45 °.

3. We get points A1, A2, A3, A4, A5, A6, A7, A8. We connect them one by one and get a regular octagon. Back

Constructing a Regular Polygon on a Side Using Rotation You can construct a regular polygon by knowing its angles. We know that the sum of the angles of a convex n-gon is 180 ° (n - 2). From this, you can calculate the angle of the polygon by dividing the sum by n. Angles Construction

Correct angle: 3-gon is 60 ° 4-gon is 90 ° 5-gon is equal to 108 ° 6-gon is 120 ° 8-gon is 135 ° 9-gon is 140 ° 10-gon is 144 ° 12-gon is 150 ° Degree measure of angles of regular triangles Back

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In 1796, one of the greatest mathematicians of all time, Karl Friedrich Gauss, showed the possibility of constructing regular n-gons if equality holds, where n is the number of angles, and k is any natural number... Thus, it turned out that within 30 it is possible to divide the circle into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30 equal parts. In 1836, Wanzel proved that regular polygons that do not satisfy this equality cannot be constructed using a ruler and compass. Gauss's theorem

Constructing a triangle Construct a circle centered at point O. Construct another circle of the same radius passing through point O.

3. Connect the centers of the circles and one of the points of their intersection, getting a regular polygon. Back Draw a triangle

Construction of a hexagon 1. Construct a circle centered at the point O. 2. Draw straight line through the center of the circle. 3. Draw an arc of a circle of the same radius centered at the point of intersection of a straight line with a circle until it intersects with a circle.

4. Draw straight lines through the center of the initial circle and the point of intersection of the arc with this circle. 5. We connect the points of intersection of all lines with the original circle and get a regular hexagon. Constructing a hexagon

Construction of a quadrilateral Let us construct a circle centered at point O. Draw 2 mutually perpendicular diameters. From the points at which the diameters touch the circle, draw other circles given radius before their intersection (circles).

Construction of a quadrilateral 4. Draw straight lines through the points of intersection of the circles. 5. We connect the points of intersection of lines and a circle and get a regular quadrilateral.

Constructing an octagon You can construct any regular polygon that has 2 times more angles than the given one. Let's build an octagon using a quadrangle. Let's connect the opposite vertices of the quadrangle. Let's draw the bisectors of the angles formed by intersecting diagonals.

4. Connect the points lying on the circle, thus obtaining a regular octagon. Constructing an octagon

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Constructing a decagon Let us construct a circle centered at point O. Draw 2 mutually perpendicular diameters. We divide the radius of the circle in half and from the point obtained on it we draw a circle passing through the point O.

Constructing a decagon 4. Draw a segment from the center of the small circle to the point at which the large circle touches its radius. 5. From the point of contact between the large circle and its radius, draw a circle so that it touches the small one.

Construction of the decagon 6. From the points of intersection of the large and the resulting circles, draw the circles constructed last time and so we will draw until the adjacent circles touch. 7. Connect the dots and get a decagon.

Constructing a pentagon To construct a regular pentagon, you need to connect one by one not all points, but through one, while constructing a regular decagon.

An approximate construction of a regular pentagon by the Dürer method. Let's construct 2 circles passing through the center of each other. We connect the centers of a straight line, getting one of the sides of the pentagon. Let's connect the points of intersection of the circles.

Approximate construction of a regular pentagon by the Dürer method 4. Draw another circle of the same radius with the center at the intersection of two other circles. 5. Let's draw 2 segments as shown in the figure.

Approximate construction of a regular pentagon by the Dürer method 6. Connect the points of contact of these segments with circles with the ends of the constructed side of the pentagon. 7. Let's finish up to a pentagon.

Approximate construction of a regular pentagon using the methods of Kovarzhik, Bion

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, install 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 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. Then, through these points, we draw the horizontal sides of the square 4-1 and 3-2 with the help of the flight tire. Then, using 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 their mutual intersection.

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. We divide the segment AO in half. To do this, we describe the arc with the radius AO from point A until it intersects 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 get the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Constructs a regular pentagon along a 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, 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.

In drafting, it is often required to construct positive polygons. So let's say positive octagons are used on road sign boards.

You will need

• - compass
• - ruler
• - pencil

Instructions

1. Let a segment be given equal to the side length of the desired octagon. It is required to build a correct octagon. In the first step, draw an isosceles triangle on the given line segment, using the line segment as the base. To do this, first build a square with a side equal to the line segment, draw diagonals in it. Now build the bisectors of the angles at the diagonals (in the figure, the bisectors are indicated in blue), a vertex is formed at the intersection of the bisectors isosceles triangle whose sides are equal to the radius of a circle circumscribed around a true octagon.

2. Construct a circle centered at the apex of the triangle. The radius of the circle is equal to the side of the triangle. Now spread the compass to a distance equal to the size of the specified segment. Draw this distance along the circle, starting at either end of the line. Combine all the resulting points into an octagon.

3. If a circle is specified in which the octagon should be inscribed, then the construction will be even simpler. Draw two centerlines, perpendicular to each other, through the center of the circle. At the intersection of the axial and the circle, there will be four vertices of the coming octagon. It remains to divide the distance between these points on the circular arc in half in order to get four more vertices.

Loyal triangle- one in which all sides have the same length. Based on this definition, the construction of a similar variety triangle a is not a difficult task.

You will need

• Ruler, sheet of lined paper, pencil

Instructions

1. Take a sheet of blank paper, lined in a box, a ruler and mark three points on the paper so that they are at an identical distance from each other (Fig. 1)

2. With the help of a ruler, combine the points marked on the sheet in steps, one by one, as shown in Figure 2.

Note!
In a true (equilateral) triangle, all angles are 60 degrees.

Helpful advice
An equilateral triangle is also isosceles. If a triangle is isosceles, then this means that 2 of its 3 sides are equal, and the third side is considered the base. Any positive triangle is isosceles, while the converse statement is not correct.

Octagon Are, in essence, two squares, offset relative to each other by 45 ° and united at the vertices by a solid line. And therefore, in order to positively depict such a geometric figure, you need to draw a square or a circle with a solid pencil hefty, according to the rules, with which to carry out subsequent actions. The presentation is focused on the side length equal to 20 cm. So, when placing the drawing, consider that the vertical and horizontal lines 20 cm long fit on a sheet of paper.

You will need

• Ruler, right triangle, protractor, pencil, compasses, sheet of paper

Instructions

1. Method 1. Draw a horizontal line with a length of 20 cm at the bottom. Then, on one side, sweep a right angle with a protractor, the one that is 90 °. The same can be done with the support of a right triangle. Draw a vertical line and sweep 20 cm. Do the same manipulations on the other side. Combine the two obtained points with a horizontal line. As a result, it turned out geometric figure- square.

2. In order to build the 2nd (offset) square, you need the center of the figure. To do this, divide each side of the square into 2 parts. First, unite the 2 points of the parallel top and bottom sides, and then the points of the sides. Draw 2 straight lines through the center of the square, perpendicular to each other. Starting from the center, measure 10 cm on new straight lines, which will result in 4 straight lines. Combine the 4 outer points obtained with each other, resulting in the 2nd square. Now combine any point from the 8 obtained angles with each other. This will draw an octagon.

3. Method 2. This will require a compass, a ruler and a protractor. From the center of the sheet with the support of a compass, draw a circle with a diameter of 20 cm (radius 10 cm). Draw a straight line through the center point. Then draw a second line perpendicular to it. The same can be done with the help of a protractor or a right triangle. As a result, the circle will be divided into 4 equal parts. Then divide each of the sections into 2 more parts. To do this, it is also allowed to use a protractor, measuring 45 ° or right triangle, the one that you attach with an acute angle of 45 ° and draw the rays. Measure 10 cm from the center on any straight line. As a result, you will get 8 "rays", which you combine with each other. The end result is an octagon.

4. Method 3. To do this, also draw a circle, draw a line through the middle. After that, take a protractor, place it in the center and measure the angles, considering that each section of the octagon has a 45 ° angle in the center. Later, on the received rays, measure the length of 10 cm and combine them with each other. Octagon ready.

Helpful advice
Make the drawing with a hard pencil, the side lines on which after that it will be easy to remove

A true octagon is a geometric figure in which every angle is 135 °, and all sides are equal to each other. This figure is hefty often used in architecture, for example, in the construction of columns, as well as in the manufacture of the STOP road sign. How do you draw a positive octagon?

You will need

• - album sheet;
• - pencil;
• - ruler;
• - compasses;
• - eraser.

Instructions

1. Draw a square first. After that, draw a circle so that the square is inside the circle. Now draw two centerlines of the square - horizontal and vertical until it intersects with the circle. Unite with straight lines the points of intersection of the axes with the circle and the points of contact of the circumscribed circle with the square. Thus, you will get the sides of the correct octagon.

2. Draw the correct octagon in a different way. Draw a circle first. Then draw a horizontal line through its center. Sweep the intersection of the rightmost border of the circle with the horizontal. This point will be the center of another circle with a radius equal to the previous shape.

3. Draw a vertical line through the intersection of the 2nd circle with the first. Place the leg of your compass at the intersection of the vertical with the horizontal and draw a small circle with a radius equal to the distance from the center of the tiny circle to the center of the starting circle.

4. Draw a straight line through two points - the center of the starting circle and the intersection of the vertical and the tiny circle. Continue it until it intersects with the line of the original shape. This will be the vertex point of the octagon. Using a compass, mark another point by drawing a circle centered at the intersection of the rightmost border of the initial circle with the horizontal and radius equal to the distance from the center to the tighter vertex of the octagon.

5. Draw a straight line through two points - the center of the starting circle and the last newly formed point. Continue in a straight line until it intersects with the boundaries of the original shape.

6. Unite with straight line segments stepwise: the point of intersection of the horizontal with the right border of the initial figure, then clockwise all the formed points, including the points of intersection of the axes with the original circle.

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