Dividing a circle into any number of equal parts

Division of a circle into three equal parts. Install a square with angles of 30 and 60 ° with a large leg parallel to one of the center lines. Along the hypotenuse from a point 1 (first division) draw a chord (Fig. 2.11, a), getting the second division - point 2. Turning the square and drawing the second chord, get the third division - point 3 (Fig. 2.11, b). By connecting points 2 and 3; 3 and 1 straight lines form an equilateral triangle.

Rice. 2.11.

a, b - c using a square; in- using a circle

The same problem can be solved using a compass. By placing the support leg of the compass at the lower or upper end of the diameter (Fig. 2.11, in) describe an arc whose radius is equal to the radius of the circle. Get the first and second divisions. The third division is at the opposite end of the diameter.

Dividing a circle into six equal parts

The compass opening is set equal to the radius R circles. From the ends of one of the diameters of the circle (from the points 1, 4 ) describe arcs (Fig. 2.12, a, b). points 1, 2, 3, 4, 5, 6 divide the circle into six equal parts. By connecting them with straight lines, they get a regular hexagon (Fig. 2.12, b).

Rice. 2.12.

The same task can be performed using a ruler and a square with angles of 30 and 60 ° (Fig. 2.13). The hypotenuse of the square must pass through the center of the circle.

Rice. 2.13.

Dividing a circle into eight equal parts

points 1, 3, 5, 7 lie at the intersection of the center lines with the circle (Fig. 2.14). Four more points are found using a square with angles of 45 °. When receiving points 2, 4, 6, 8 the hypotenuse of a square passes through the center of the circle.

Rice. 2.14.

Dividing a circle into any number of equal parts

To divide a circle into any number of equal parts, use the coefficients given in Table. 2.1.

Length l chord, which is laid on a given circle, is determined by the formula l = dk, where l- chord length; d is the diameter of the given circle; k- coefficient determined from Table. 1.2.

Table 2.1

Coefficients for dividing circles

To divide a circle of a given diameter of 90 mm, for example, into 14 parts, proceed as follows.

In the first column of Table. 2.1 find the number of divisions P, those. 14. From the second column write out the coefficient k, corresponding to the number of divisions P. In this case, it is equal to 0.22252. The diameter of a given circle is multiplied by a factor and the length of the chord is obtained l=dk= 90 0.22252 = 0.22 mm. The resulting length of the chord is set aside with a measuring compass 14 times on a given circle.

Finding the center of the arc and determining the size of the radius

An arc of a circle is given, the center and radius of which are unknown.

To determine them, you need to draw two non-parallel chords (Fig. 2.15, a) and set up perpendiculars to the midpoints of the chords (Fig. 2.15, b). Center O arc is at the intersection of these perpendiculars.

Rice. 2.15.

Pairings

When performing engineering drawings, as well as when marking workpieces in production, it is often necessary to smoothly connect straight lines with arcs of circles or an arc of a circle with arcs of other circles, i.e. perform pairing.

Pairing called a smooth transition of a straight line into an arc of a circle or one arc into another.

To build mates, you need to know the value of the radius of the mates, find the centers from which the arcs are drawn, i.e. interface centers(Fig. 2.16). Then you need to find the points at which one line passes into another, i.e. connection points. When constructing a drawing, mating lines must be brought exactly to these points. The point of conjugation of the arc of a circle and a straight line lies on a perpendicular lowered from the center of the arc to the mating line (Fig. 2.17, a), or on a line connecting the centers of mating arcs (Fig. 2.17, b). Therefore, to construct any conjugation by an arc of a given radius, you need to find interface center and point (points) conjugation.

Rice. 2.16.

Rice. 2.17.

The conjugation of two intersecting lines by an arc of a given radius. Given straight lines intersecting at right, acute and obtuse angles (Fig. 2.18, a). It is necessary to construct conjugations of these lines by an arc of a given radius R.

Rice. 2.18.

For all three cases, the following construction can be applied.

1. Find a point O- the center of the mate, which must lie at a distance R from the sides of the corner, i.e. at the point of intersection of lines passing parallel to the sides of the angle at a distance R from them (Fig. 2.18, b).

To draw straight lines parallel to the sides of an angle, from arbitrary points taken on straight lines, with a compass solution equal to R, make serifs and draw tangents to them (Fig. 2.18, b).

• 2. Find the junction points (Fig. 2.18, c). For this, from the point O drop perpendiculars to given lines.
• 3. From point O, as from the center, describe an arc of a given radius R between junction points (Fig. 2.18, c).

To divide a circle in half, it is enough to draw any diameter of it. Two mutually perpendicular diameters will divide the circle into four equal parts (Figure 28, a). By dividing each fourth part in half, eighths are obtained, and with further division - sixteenth, thirty-second parts, etc. (Figure 28, b). If connect the division points with straight lines, then you can get the sides of a regular inscribed square ( a 4 ), octagon ( a 8 ) and t . d. (Figure 28, c).

Figure 28

Dividing the circle into 3, 6, 12, etc., equal parts, as well as construction of the corresponding regular inscribed polygons carried out as follows. Two mutually perpendicular diameters are drawn in a circle 1–2 and 3–4 (Figure 29 a). From points 1 and 2 how to describe arcs from the centers with the radius of a circle R before intersecting with it at points A, B, C and D . points A , B , 1, C, D and 2 divide the circle into six equal parts. The same points, taken through one, will divide the circle into three equal parts (Figure 29, b). To divide the circle into 12 equal parts, describe two more arcs with a radius of a circle from points 3 and 4 (Figure 29, c).

Figure 29

You can also build a regular inscribed triangle, hexagon, etc. using a ruler and a square at 30 and 60 °. Figure 30 shows a similar construction for an inscribed triangle.

Figure 30

Dividing a circle into seven equal parts and the construction of a regular inscribed heptagon (Figure 31) is performed using half the side of the inscribed triangle, approximately equal side inscribed heptagon.

Figure 31

To divide a circle into five or ten equal parts two mutually perpendicular diameters are carried out (Figure 32, a). Radius OA divide in half and, having received a point AT , describe from it an arc with a radius R=BC until it intersects at a point D with a horizontal diameter. Distance between points C and D equal to the length of a side of a regular inscribed pentagon ( a 5 ), and the segment OD equal to the length of a side of a regular inscribed decagon ( a 10 ). The division of the circle into five and ten equal parts, as well as the construction of the inscribed regular pentagon and decagon are shown in Figure 32, b. An example of the use of dividing a circle into five parts is five pointed star(Figure 32, c).

Figure 32

Figure 33 shows a general method for the approximate division of a circle into equal parts . Let it be required to divide the circle into nine equal parts. Two mutually perpendicular diameters and a vertical diameter are drawn in a circle. AB divided into nine equal parts using an auxiliary straight line (Figure 33, a). From a point B describe an arc with a radius R = AB, and at its intersection with the continuation of the horizontal diameter, points are obtained FROM and D . From points C and D through even or odd diameter division points AB conduct rays. The intersection points of the rays with the circle will divide it into nine equal parts (Figure 33, b).

Today in the post I post several pictures of ships and diagrams for them for embroidery with isothread (pictures are clickable).

Initially, the second sailboat was made on carnations. And since the carnation has a certain thickness, it turns out that two threads depart from each. Plus, layering one sail on the second. As a result, a certain effect of splitting the image appears in the eyes. If you embroider the ship on cardboard, I think it will look more attractive.
The second and third boats are somewhat easier to embroider than the first. Each of the sails has a central point (on the underside of the sail) from which rays extend to points along the perimeter of the sail.
Joke:
- Do you have threads?
- There is.
- And the harsh ones?
- It's just a nightmare! I'm afraid to come!

My first debut Master Class. Hopefully not the last. We will embroider a peacock. Product diagram.When marking the puncture sites, pay attention Special attention so that in closed circuits there are even number.The basis of the picture is dense cardboard(I took brown with a density of 300 g / m2, you can try it on black, then the colors will look even brighter), better dyed on both sides(for the people of Kiev - I took it in the stationery department at the Central Department Store on Khreshchatyk). Threads- floss (of any manufacturer, I had DMC), in one thread, i.e. we unwind the bundles into individual fibers. Embroidery consists of three layers thread. First we embroider the first layer in feathers on the peacock's head, the wing (light blue thread color), as well as dark blue circles of the tail using the flooring method. The first layer of the body is embroidered with chords with variable pitch, trying to make the threads run tangentially to the contour of the wing. Then we embroider twigs (serpentine seam, mustard-colored threads), leaves (first dark green, then the rest ...

Dividing a circle into four equal parts and constructing a regular inscribed quadrilateral(Fig. 6).

Two mutually perpendicular center lines divide the circle into four equal parts. By connecting the points of intersection of these lines with the circle with straight lines, a regular inscribed quadrilateral is obtained.

Dividing a circle into eight equal parts and constructing a regular inscribed octagon(Fig. 7).

The division of the circle into eight equal parts is carried out using a compass as follows.

From points 1 and 3 (the points of intersection of the center lines with the circle) with an arbitrary radius R, arcs are drawn to mutual intersection, with the same radius from point 5, a notch is made on the arc drawn from point 3.

Straight lines are drawn through the points of intersection of the serifs and the center of the circle until they intersect with the circle at points 2, 4, 6, 8.

If the obtained eight points are connected in series by straight lines, then a regular inscribed octagon will be obtained.

Dividing a circle into three equal parts and constructing a regular inscribed triangle(Fig. 8).

Option 1.

When dividing the circle with a compass into three equal parts from any point on the circle, for example, point A of the intersection of the center lines with the circle, draw an arc with a radius R equal to the radius of the circle, get points 2 and 3. The third division point (point 1) will be located at the opposite end of the diameter , passing through point A. by successively connecting points 1, 2 and 3, a regular inscribed triangle is obtained.

Option 2.

When constructing a regular inscribed triangle, if one of its vertices, for example, point 1, is given, point A is found. To do this, through given point carry out the diameter (Fig. 8). Point A will be at the opposite end of this diameter. Then an arc is drawn with a radius R equal to the radius of the given circle, points 2 and 3 are obtained.

Dividing a circle into six equal parts and constructing a regular inscribed hexagon(Fig. 9).

When dividing the circle into six equal parts using a compass from two ends of the same diameter with a radius equal to the radius of the given circle, arcs are drawn until they intersect with the circle at points 2, 6 and 3, 5. Connecting the points obtained in succession, a regular inscribed hexagon is obtained.

Dividing a circle into twelve equal parts and constructing a regular inscribed dodecagon(Fig. 10).

When dividing a circle with a compass from the four ends of two mutually perpendicular diameters of the circle, an arc is drawn with a radius equal to the radius of the given circle, until it intersects with the circle (Fig. 10). By connecting the intersection points obtained in succession, a regular inscribed dodecagon is obtained.

Dividing a circle into five equal parts and constructing a regular inscribed pentagon ( Fig.11).

When dividing a circle with a compass, half of any diameter (radius) is divided in half, point A is obtained. From point A, as from the center, an arc is drawn with a radius equal to the distance from point A to point 1, until it intersects with the second half of this diameter at point B. Segment 1B is equal to the chord subtending the arc, the length of which is equal to 1/5 of the circumference. Making serifs on a circle with a radius R1 equal to the segment 1B, the circle is divided into five equal parts. The starting point A is chosen depending on the location of the pentagon.

Points 2 and 5 are built from point 1, then point 3 is built from point 2, and point 4 is built from point 5. The distance from point 3 to point 4 is checked with a compass; if the distance between points 3 and 4 is equal to the segment 1B, then the constructions were performed exactly.

It is impossible to perform serifs sequentially, in one direction, since measurement errors accumulate and the last side of the pentagon turns out to be skewed. Consistently connecting the found points, a regular inscribed pentagon is obtained.

Dividing a circle into ten equal parts and constructing a regular inscribed decagon(Fig. 12).

The division of the circle into ten equal parts is performed similarly to the division of the circle into five equal parts (Fig. 11), but first the circle is divided into five equal parts, starting from point 1, and then from point 6, located at the opposite end of the diameter. By connecting all the points in series, a regular inscribed decagon is obtained.

Dividing a circle into seven equal parts and constructing a regular inscribed heptagon(Fig. 13).

From any point of the circle, for example, point A, an arc is drawn with a radius of a given circle until it intersects with a circle at points B and D of a straight line.

Half of the resulting segment (in this case, segment BC) will be equal to the chord that subtends the arc, which is 1/7 of the circumference. With a radius equal to the segment BC, serifs are made on the circle in the sequence shown when constructing a regular pentagon. By connecting all the points in series, a regular inscribed heptagon is obtained.

Dividing the circle into fourteen equal parts and constructing a regular inscribed fourteen-angle (Fig. 14).

The division of the circle into fourteen equal parts is performed similarly to the division of the circle into seven equal parts (Fig. 13), but first the circle is divided into seven equal parts, starting from point 1, and then from point 8, located at the opposite end of the diameter. By connecting all the points in series, they get a regular inscribed tetragon.

During repairs, you often have to deal with circles, especially if you want to create interesting and original decor elements. It is also often necessary to divide them into equal parts. There are several methods to do this. For example, you can draw a regular polygon or use tools known to everyone since school. So, in order to divide the circle into equal parts, you will need the circle itself with a well-defined center, a pencil, a protractor, as well as a ruler and a compass.

Dividing a circle with a protractor

Dividing a circle into equal parts using the above tool is perhaps the easiest. We know that a circle is 360 degrees. Dividing this value by right amount parts, you can find out how much each part will take (see photo).

Further, starting from any point, you can make notes corresponding to the calculations. This method is good when the circle needs to be divided by 5, 7, 9, etc. parts. For example, if the figure needs to be divided into 9 parts, the marks will be at 0, 40, 80, 120, 160, 200, 240, 280 and 320 degrees.

Division into 3 and 6 parts

To correctly divide the circle into 6 parts, you can use the property of a regular hexagon, i.e. its longest diagonal must be twice the length of its side. To begin with, the compass must be stretched to a length equal to the radius of the figure. Next, leaving one of the legs of the tool at any point on the circle, the second one needs to be marked, after which, repeating the manipulations, it will turn out to make six points, connecting which you can get a hexagon (see photo).

By connecting the vertices of the figure through one, you can get a regular triangle, and accordingly the figure can be divided into 3 equal parts, and by connecting all the vertices and drawing diagonals through them, you can divide the figure into 6 parts.

Division into 4 and 8 parts

If the circle needs to be divided into 4 equal parts, first of all, it is necessary to draw the diameter of the figure. This will allow you to get two of the required four points at once. Next, you need to take a compass, stretch its legs along the diameter, after which one of them should be left at one of the ends of the diameter, and the other should be made notches outside the circle from the bottom and top (see photo).

The same must be done for the other end of the diameter. After that, the points obtained outside the circle are connected with a ruler and a pencil. The resulting line will be the second diameter, which will be clearly perpendicular to the first, as a result of which the figure will be divided into 4 parts. In order to obtain, for example, 8 equal parts, the resulting right angles can be divided in half and diagonals drawn through them.