Dividing the circumference to the equal parts using a circulation and a ruler. Dividing the circle to six equal parts and the construction of the correct inscribed hexagon

The division of the circumference into six equal parts and the construction of the correct inscribed hexagon is performed using a square with angle 30, 60 and 90 º and / or circula. When dividing a circle to six equal parts with a circulation of two ends of one diameter by a radius equal to a radius of this circle, arcs are carried out to crossing with a circle at points 2, 6 and 3, 5 (Fig. 2.24). Consistently by connecting the points obtained, the correct inscribed hexagon is obtained.

Figure 2.24

When dividing the circumference with a circulation of four ends of two mutually perpendicular diameters of the circle is carried out by a radius equal to the radius of this circle, arcs up to the intersection with a circle (Fig. 2.25). By connecting the obtained points, a twelve brittle is obtained.

Figure 2.25

2.2.5 division of a circle for five and ten equal parts
and the construction of the right inscribed pentagon and a decidagon

The division of the circumference is five and ten equal parts and the construction of the correct entered pentagon and a decidagon is shown in Fig. 2.26.

Figure 2.26

Half of any diameter (radius) are divided by half (Fig. 2.26 a), the point of A.Is points A, as from the center, carry out an arc with a radius equal to the distance from the point of the ADO point 1 to the intersection from the second half of this diameter, at the point in ( Fig. 2.26 B ). The segment is 1 erased by the chore, a tightening arc, the length of which is 1/5 of the length of the circle. Making sneakers on the circle (Fig. 2.26, in ) radius TOAn equal to the segment 1B is divided into five equal parts. The starting point 1 is chosen depending on the location of the pentagon. From point 1, they build points 2 and 5 (Fig. 2.26, B), then from point 2 build a point 3, and from point 5 build a point 4. The distance from point 3 to point 4 is checked with a circulation. If the distance between points 3 and 4 is equal to the segment 1B, then the construction was performed exactly. You can not perform serifs sequentially, in one direction, since the error rafting and the last side of the pentagon turns out to be powered. Consistently by connecting the found points, a pentagon is obtained (Fig. 2.26, d).

The division of the circumference by ten equal parts is performed similarly to dividing the circle to five equal parts (Fig. 2.26), but first divide the circle for five parts, starting construction from point 1, and then from point 6, located at the opposite end of diameter (Fig. 2.27, but). Connecting successively all points, get the correct inscribed decidagon (Fig. 2.27, b).

Figure 2.27

2.2.6 dividing the circumference by seven and fourteen equal
parts and construction of the right inscribed semi-broth and
Fourteentian

The division of the circumference by seven and fourteen equal parts and the construction of the correct inscribed sevenfound and the fourteenty brittle is shown in Fig. 2.28 and 2.29.

From anywhere around the circumference, for example, points a , the radius of a given circle is carried out with an arc (Fig. 2.28, and ) before the intersection with a circle at points in and d . Connect the dots vinphum. Half of the resulting segment (in this case, the segment of the Sun) will be equal to chord, which tightens the arc, which is 1/7 of the circumference length. A radius equal to the segment of the Sun is made of serifs on the circle in the sequence shown in Fig. 2.28, B. . By connecting all the points sequentially, the correct inscribed sevenfoon is obtained (Fig. 2.28, B).

The division of the circle to fourteen equal parts is performed by dividing the circle to seven equal parts twice from two points (Fig. 2.29, a).

Figure 2.28.

First, the circle is divided into seven equal parts from point 1, then the same construction is executed from point 8 . The constructed points are connected in sequentially straight lines and get the correct part of the fourteenty bron (Fig. 2.29, b).

Figure 2.29.

Building an ellipse

The image of the circle in a rectangular isometric projection in all three planes of projections is the same in the form of ellipses.

The direction of the small axis of the ellipse coincides with the direction of the axonometric axis, perpendicular to the plane of the projections in which the circle is presented.

When constructing an ellipse depicting a circle of a small diameter, it suffices to build eight points belonging to the ellipse (Fig. 2.30). Four of them are the ends of the ellipse axes (A, B, C, D), and the four others (N 1, N 2, N 3, N 4) are located on direct, parallel axonometric axes, at a distance equal to the radius of the mapped circle from the center ellipse.

When performing graphic work, it is necessary to solve many construction tasks. The most occurring tasks are dividing the segments of straight, angles and circles to equal parts, the construction of various conjugations.

Dividing circle on equal parts with a circulation

Using the radius, it is easy to divide the circle and 3, 5, 6, 7, 8, 12 of equal sections.

Dividing circle to four equal parts.

Bunctuous center lines, carried out perpendicular to one other, divide the circle to four equal parts. Connecting their ends consistently, we get the right quadrangle (Fig. 1) .

Fig.1 Dividing circle to 4 equal parts.

Dividing circle to eight equal parts.

To divide the circle to eight equal parts, arcs equal to the fourth part of the circle are divided by half. For this, two points that limit the quarter of the arc, as from the centers of the circle, serves serifs from its limits. The obtained points are connected to the center of circles and at the intersection of them with the line of the circle receive points that divide the fourth sections in half, that is, they receive eight equal sections of the circle (Fig. 2 ).

Fig.2. Dividing circle on 8 equal parts.

Dividing a circle to sixteen equal parts.

Dividing an arc with a circle equal to 1/8, into two equal parts, we will apply a sector to a circle. By connecting all the serifs, the sections of the straight lines, we obtain the right sixteen-carton.

Fig.3. Dividing circle on 16 equal parts.

Dividing circle into three equal parts.

To split the circle of radius R into 3 equal parts, from the point of intersection of the center line with a circle (for example, from point A) are described as an additional arc of R. R. Points 2 and 3. Points 1, 2, 3 divide the circle to three equal Parts.

Fig. four. Dividing circle to 3 equal parts.

Dividing circle into six equal parts. The side of the right hexagon entered into the circle is equal to the radius of the circumference (Fig. 5.).

To divide the circle to six equal parts, it is necessary from the points 1 and 4 intersection of the center line with a circle to make two serfs in a circle by a radius R.equal to the radius of the circle. By connecting the points received by sections of the straight lines, we obtain the correct hexagon.

Fig. 5. division of a circle on 6 equal parts

Dividing circle on twelve equal parts.

To divide the circle on twelve equal parts, it is necessary to divide the circle to four parts mutually perpendicular diameters. Taking the intersection points of diameters with a circle BUT , IN, FROM, D. for centers, the amount of radius is carried out four arcs before crossing the circle. Points received 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and points BUT , IN, FROM, D. separate a circle on twelve equal parts (Fig. 6).

Fig. 6. division of a circle at 12 equal parts

Dividing circle for five equal parts

From the point BUT We carry out the arc with the same radius as the radius of the circle to the intersection with the circle - we get a point IN. Lowering perpendicular from this point - get a point FROMPoints FROM - middle of the circle radius, like from the center, the arc of the radius Cd. Let's drink in diameter, get a point E.. Section DE. It is equal to the length of the side of the proper proper pentagon. By making a radius DE. Sexing on the circle, we obtain the point of dividing the circle to five equal parts.

Fig. 7. division of a circle on 5 equal parts

Dividing circumference by ten equal parts

Dividing the circle to five equal parts, it is easy to divide the circle and on 10 equal parts. After spending direct from the resulting points through the center of the circumference to the opposite sides of the circle - we get another 5 points.

Fig. 8. division of a circle at 10 equal parts

Dividing circle to seven equal parts

To divide the circle of radius R. on 7 equal parts, from the intersection point of the center line with a circle (for example, from the point BUT) describe how from the center an additional arc similar radius R. - Get a point IN. Holding perpendicular from the point IN - We get a point FROM.Section Sun It is equal to the length of the side of the properly seven-angry.

Fig. 9. division of a circle on 7 equal parts

Dividing the circumference to four equal parts and the construction of the right inscribed quadrangle (Fig. 6).

Two mutually perpendicular centers divide the circle to four equal parts. By connecting the intersection points of these lines with a circle straight, get the right fetched quadrangle.

Dividing the circle to eight equal parts and the construction of the right inscribed octagon (Fig. 7).

The division of the circle on eight equal parts is performed using a circulation as follows.

From points 1 and 3 (points of intersection of centered lines with a circle) an arbitrary radius of R conduct arcs to a mutual intersection, the same radius from point 5 make a serif on an arc conducted from point 3.

Through the points of intersection of sneakers and the center of the circumference, direct lines are carried out to the intersection with a circle at points 2, 4, 6, 8.

If the eight points obtained are connected consistently with straight lines, it will turn out the correct entitled octagon.

Dividing the circumference to three equal parts and the construction of the correct part of the triangle(Fig. 8).

Option 1.

When dividing the circular circle to three equal parts from any point of the circle, for example, the points and the intersection of centered lines with a circle carry out an arc with a radius R, equal to the circle radius, obtain points 2 and 3. The third division point (point 1) will be at the opposite end of the diameter passing through the point A. Sequentially connecting points 1, 2 and 3, get the correct inscribed triangle.

Option 2.

When constructing the correct inscribed triangle, if one of its vertices is specified, for example, point 1, they find a point A. For this, a diameter is carried out through a specified point (Fig. 8). Point A will be at the opposite end of this diameter. Then he is carried out an arc with a radius R, equal to the radius of this circle, obtain points 2 and 3.

Dividing the circle to six equal parts and the construction of the correct inscribed hexagon (Fig. 9).

When dividing the circle to six equal parts with a circulation of two ends of one diameter with a radius equal to a radius of a given circle, arcs are carried out before crossing the circle at points 2, 6 and 3, 5. Connecting the points received sequentially, the correct hexagon is obtained.

Dividing the circumference to twelve equal parts and the construction of the right inscribed duodenum (Fig.10).

When dividing the circle of the circulation of the four ends of two mutually perpendicular diameters of the circle is carried out by a radius equal to the radius of this circumference, arcs up to the intersection with a circle (Fig. 10). By connecting consistently the resulting intersection points are obtained by the correct inscribed twelve bron.

Dividing the circumference of five equal parts and the construction of the correct fetched pentagon (fig.11).

When dividing the circle with a circular half of any diameter (radius), they are divided by half, get a point of A. from point A, as from the center, carry out an arc with a radius equal to the distance from point A to point 1, to the intersection from the second half of this diameter at the point V. Cut 1B is equal to the chopping arc, the length of which is 1/5 of the circumference length. Making serifs on the circle with radius R1, equal to the segment 1B, divide the circle to five equal parts. The starting point A is chosen depending on the location of the pentagon.

From point 1, the points 2 and 5 are built, then the point 2 is built from point 2, and the point 5 build a point 4. The distance from point 3 to point 4 is checked with a circulation; If the distance between points 3 and 4 is equal to the segment 1B, then the construction was performed exactly.

You can not perform serifs sequentially, in one direction, since the accumulation of measurement errors is accumulated and the last side of the pentagon turns out to be powered. Consistently by connecting the found points, get the correct part of the pentagon.

Dividing the circumference of ten equal parts and the construction of the right inscribed decidagon(Fig.12).

The division of the circle at ten equal parts is performed similarly to the division of the circle to five equal parts (Fig. 11), but first divide the circle to five equal parts, starting to build from point 1, and then from point 6, located at the opposite end of the diameter. By connecting the sequentially all points, the proper enclosed decidagon is obtained.

Dividing the circumference to seven equal parts and the construction of the right inscribed semi-broth (Fig.13).

From any point of the circle, for example, points A, a radius of a given circumference, an arc is carried out to the intersection with a circle at points B and D direct.

Half of the resulting segment (in this case, the segment of the Sun) will be equal to chord, which tightens the arc, which is 1/7 of the circumference length. A radius equal to the segment of the aircraft, there are serifs on the circle in the sequence shown in constructing the correct pentagon. Connecting successively all points, get the right inscribed sevenfoon.

Dividing the circumference to fourteen equal parts and the construction of the correct part of the fourteenthist (Fig.14).

The division of the circumference by fourteen equal parts is performed similarly to the division of the circle at seven equal parts (Fig. 13), but first divide the circle to seven equal parts, starting constructing from point 1, and then from point 8, located at the opposite end of the diameter. Connecting successively all points, get the correct fourteenty trigger.

Dividing circle on equal parts

Division of 3 parts (Fig. 12, but). From the end of the diameter of the circumference, an arc is carried out by a radius R.equal to the radius of the circle. The arc forms two necessary points on the circle. The third point is at the opposite end of the diameter.

Division of 4 and 8 parts. When dividing a circle on 4 parts, a circulation and a ruler will be helped, with which it is necessary to carry out two mutually perpendicular diameters (Fig. 12, b.). If you spend one diameter and from one end to describe the arc is somewhat big than the radius R., and from the opposite end of the diameter to carry out another arc of the same radius, then by connecting the points of their intersection of the straight line (which will be held through the center), we obtain the second diameter perpendicular to the first. The intersection points perpendicular diameters with a circle divide it into 4 equal parts.

To divide the circle on 8 equal parts (Fig. 12, in) It is necessary to construct two pairs of mutually perpendicular diameters.

Fig. 12. Dividing circle to equal parts: but - for three parts; b. - for four parts; in - for eight parts; g. - for five parts (1st way); d. - for five parts (2nd method); e. - by six parts; j. - Seven parts.

5-piece division. The division of the circle on 5 parts can be performed in several ways. First method (Fig. 12, g.) implies the use of a circulation and a ruler. First, a known method must be carried out two mutually perpendicular diameters. After that, radius R. It is necessary to divide in half: from the extreme point of intersection of the horizontal diameter it is necessary to carry out an arc of radius R. And after two points formed when crossing this arc with a circle, spend a straight line - it will split the horizontal line of the radius R. in half. From the division point (? R.) conduct an arc with a radius r. (equal distance from the point? R. to the intersection point of the circle with a vertical diameter). This arc cross the second half of the horizontal diameter at the point FROM. Cut equal to the distance from the point FROM The point of intersection of the circle with a vertical diameter will correspond to the part of the sought-in pentagon circumference. It is necessary to establish a circuit for a value equal to the length of this segment, and from the upper point of crossing the circle with a vertical diameter to carry out an arc of a given radius - the point of its intersection with the circle will be the next peak of the pentagon. From the found vertex you need to spend another arc of a given radius - it will be the third top of the pentagon, from which, in turn, it will be necessary to carry out the following arc, and so far the circle is not divided into 5 equal parts. If after this is to carry out the next five arcs of a given radius, but starting from the lower point of intersection of the circle with a vertical diameter, the circle is divided into 10 equal parts. In addition, in fig. 12, g., segment allocated SO On the horizontal diameter corresponding to the 1/10 circumference, that is, if there are 10 arcs on the circle to carry out 10 arcs by a radius corresponding to the length of the segment SO, The circle is also divided into 10 equal parts.

In the second method (Fig. 12, d.) On the diameter of the circle using the already known reception, it is necessary to find a point that will divide the radius R. in half. From this point spend a straight line to the intersection with the end of the diameter (points FROM). Then from the point R./ 2 conduct an arc with a radius equal to? R., before its intersection with the conducted line at the point E.. Next Circle from point FROM conduct an arc with a radius equal to the segment CE, before its intersection with a circle at points BUT and IN. Section AU - Grand Pentagon. Now it remains to spend out of points BUT and IN arc radius equal to the magnitude of the segment AUTo sequentially divide the circle on 5 parts.

There is also a way of dividing the circle on 5 parts using the transport. To radius R. Circle must be attached to the transportation, construct a central angle of 72 ° (360: 5 \u003d 72) and spend from the center a direct line to the point of its intersection with a circle. The resulting point must be connected to the radius intersection point. R. On the circle - this segment is a pentagon side. After conducting from both arc points with a radius corresponding to the length of this segment, a circle on 5 parts can be divided.

Division of 6 and 12 parts (Fig. 12, e.). From the points of crossing the circle with a vertical diameter, two arcs are carried out, the radius of which is equal to the radius of the circle. Crossing arcs on the circle forms points that are consistently connected by chords. As a result, the hexagonist is part of the circle inscribed. To divide the circle on 12 parts, the same construction is made, but only on two mutually perpendicular diameters.

Division into 7 pieces (Fig. 12, j.). From the end of any diameter, auxiliary arc radius is carried out R.. Through the points of its intersection with a circle, chord is carried out, equal to the side of the correctly inscribed triangle (as in Fig. 12, but). Half chords equates to the side inscribed in the circumference of the sevenginous. Now it is enough to sequentially postpone the circumference of several arcs with a radius equal to half of the chord to divide the circle on 7 parts.

Division on any number of parts (Fig. 13). In this case, the circle is divided into 9 parts.

Through the center of the circle, two mutually perpendicular straight lines are carried out. One of the diameters, for example CDThe line is divided into the desired number of equal parts (in this case 9), the points are numbered. Next from the point D. conduct an arc with a radius equal to the diameter of this circle (2 R.), before intersection with perpendicular direct AU. From the intersection points BUT and IN Rays are carried out, but so that they passed only through even or only through odd (as in this case) the rooms. When crossing with a circle, the rays form points that divide the circle to the desired number of parts (in this case 9).

Fig. 13. Dividing the circle on any given number of parts.

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It turns out too soft soap, disintegrating on the part with cutting if the soap during cutting disintegrates into parts and at the same time it is also very soft, oily, but you did everything right and on the right recipe, your soap, most likely, could not pass the gel phase. For solutions

To the question how to divide the circle into three equal parts of the circulation)? Tell me this please !! Posted by the author Embassy The best answer is
_______
Let it be given a range of Radius R. It is necessary to divide it into three equal parts with the help of a circulation. Open the circus on the magnitude of the radius of the circle. You can use the ruler, but you can put the circulation needle to the center of the circle, and take the leg to the link describing the circle. The line in any case is still useful later.
Install the circulation needle in an arbitrary location on the circle describing the circle, and the giffyness draw a small arc crossing the outer circle contour. Then install the needle of the circulation in the found point link and once again carry out the arc by the same radius (equal to the radius of the circle).
Repeat these actions until the next intersection point coincides with the very first. You will receive six reference to the circles located at equal intervals. It remains to choose three points through one and line to connect them with the center of the circle, and you will get a divided by a soul circle.
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Circle can be divided into three parts, if using a circulation, from the intersection point of a direct, conducted through the center of the circle o, make a circuit of the serif b and C on the line of the circumference of the value equal to the radius of this circle.
Thus, two desired points will be found, and the third is the opposite point A, where the circle and direct are intersect.
Next, if necessary, with a ruler and pencil

you can draw a built-in triangle.

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For marking into three parts, we use the circle radius.

I turn the circular on the contrary. The needle is installed on
crossing the axial line with a circle, and the stylus to the center. Outlining
arc crossing circle.

The intersection places and will be the vertices of the triangle.