Division of a circle into equal parts. Dividing a circle into equal parts and constructing regular inscribed polygons

A circle is a closed curved line, each point of which is located at the same distance from one point O, called the center.

Straight lines connecting any point on a circle with its center are called radii R.

A line AB connecting two points of a circle and passing through its center O is called diameter D.

The parts of the circles are called arcs.

A line CD joining two points on a circle is called chord.

A line MN that has only one point in common with a circle is called tangent.

The part of a circle bounded by a chord CD and an arc is called segment.

The part of a circle bounded by two radii and an arc is called sector.

Two mutually perpendicular horizontal and vertical lines intersecting at the center of a circle are called circle axes.

The angle formed by two radii of KOA is called central corner.

Two mutually perpendicular radius make an angle of 90 0 and limit 1/4 of the circle.

Division of a circle into parts

We draw a circle with horizontal and vertical axes that divide it into 4 equal parts. Drawn with a compass or square at 45 0, two mutually perpendicular lines divide the circle into 8 equal parts.

Division of a circle into 3 and 6 equal parts (multiples of 3 by three)

To divide the circle into 3, 6 and a multiple of them, we draw a circle of a given radius and the corresponding axes. The division can be started from the point of intersection of the horizontal or vertical axis with the circle. The specified radius of the circle is successively postponed 6 times. Then the obtained points on the circle are successively connected by straight lines and form a regular inscribed hexagon. Connecting points through one gives an equilateral triangle, and dividing the circle into three equal parts.

The construction of a regular pentagon is performed as follows. We draw two mutually perpendicular axes of the circle equal to the diameter of the circle. Divide the right half of the horizontal diameter in half using the arc R1. From the obtained point "a" in the middle of this segment with radius R2, we draw an arc of a circle until it intersects with the horizontal diameter at point "b". Radius R3 from point "1" draw an arc of a circle to the intersection with a given circle (point 5) and get the side of a regular pentagon. The "b-O" distance gives the side of a regular decagon.

Dividing a circle into N-th number of identical parts (building a regular polygon with N sides)

It is performed as follows. We draw horizontal and vertical mutually perpendicular axes of the circle. From the top point "1" of the circle we draw a straight line at an arbitrary angle to the vertical axis. On it we set aside equal segments of arbitrary length, the number of which is equal to the number of parts into which we divide the given circle, for example 9. We connect the end of the last segment with the lower point of the vertical diameter. We draw lines parallel to the obtained one from the ends of the segments set aside to the intersection with the vertical diameter, thus dividing the vertical diameter of the given circle into a given number of parts. With a radius equal to the diameter of the circle, from the lower point of the vertical axis we draw an arc MN until it intersects with the continuation of the horizontal axis of the circle. From points M and N we draw rays through even (or odd) division points of the vertical diameter until they intersect with the circle. The resulting segments of the circle will be the desired ones, because points 1, 2, …. 9 divide the circle into 9 (N) equal parts.

To find the center of an arc of a circle, you need to perform the following constructions: on this arc, mark four arbitrary points A, B, C, D and connect them in pairs with chords AB and CD. We divide each of the chords in half with the help of a compass, thus obtaining a perpendicular passing through the middle of the corresponding chord. The mutual intersection of these perpendiculars gives the center of the given arc and the circle corresponding to it.

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.

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. At 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 ...

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 IN , 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).

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, but), 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, but) and set up perpendiculars to the midpoints of the chords (Fig. 2.15, b). Centre ABOUT arc is at the intersection of these perpendiculars.

Rice. 2.15.

Pairings

When performing machine-building 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 the line lies on the perpendicular, lowered from the center of the arc to the conjugate line (Fig. 2.17, but), 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, but). 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 ABOUT- 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 ABOUT 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).