How to divide a circle into 3 equal parts. Dividing a circle into equal parts (how to divide)

It can be divided in two ways. For one of them you will need a compass and a ruler, and for the second one you will need a ruler and a protractor. Which option is preferable is up to you.

You will need

• - compasses
• - ruler
• - protractor

Instruction

Let a circle of radius R be given. We must divide it into three equal parts using a compass. Expand the compass by the radius of the circle. You can use a ruler in this case, or you can put the compass needle in the center of the circle, and move the leg to the circle describing the circle. The ruler will still come in handy later anyway. Set the compass needle in an arbitrary place on the circle describing the circle, and with the stylus draw a small arc that intersects the outer contour of the circle. Then set the compass needle to the found intersection point and once again draw an arc with the same radius (equal to the radius of the circle). Repeat these steps until the next intersection point matches the very first one. You will get six points on the circle spaced at regular intervals. It remains to select three points through one and connect them with a ruler to the center of the circle, and you will get a circle divided into three.

To divide a circle into three parts using a protractor, it is enough to remember that a full rotation around its axis is 360 ° -. Then the angle corresponding to one third of the circle is 360°-/3 = 120°-. Now set aside three times an angle of 120 ° - on the outside of the circle and connect the resulting points on the circle with the center.

note

If you connect the points not to the center, but to each other, you will get an equilateral triangle.

The method described in the first step also allows you to get the division of the circle into six equal parts.

And the construction of regular inscribed polygons

Dividing the circle into 3, 6 and 12 equal parts. Construction of a regular inscribed triangle, hexagon and dodecagon.

To construct a regular inscribed triangle, it is necessary from a point BUT the intersection of the center line with the circle set aside a size equal to the radius R, to one side and the other. We get vertices 1 and 2( rice. 26, a). Vertex 3 lies on the opposite point BUT end of diameter.

1/3 1/6 1/12

a B C)

Rice. 26

The side of the hexagon is equal to the radius of the circle. The division into 6 parts is shown in fig. 26, b.

In order to divide the circle into 12 parts, it is necessary to set aside a size equal to the radius on the circles in one direction and the other from four centers (Fig. 26, in).

Dividing the circle into 4 and 8

inscribed quadrilateral and octagon.

Rice. 27

The circle is divided into 4 parts by two mutually perpendicular center lines. To divide into 8 parts, an arc equal to a quarter of a circle must be divided in half ( Fig.27.)

Dividing the circle into 5 and 10 equal parts. Building the right

inscribed pentagon and decagon.

1/5 1/10

a) b)

Rice. 28

Half of any diameter (radius) is divided in half ( rice. 28, a), get a point N. From a point N, as from the center, draw an arc with a radius R1, equal to the distance from the point N to the point BUT, until it intersects with the second half of this diameter, at the point R. Line segment AR equal to a chord subtending an arc whose length is 1/5 of the circumference. Making serifs on a circle with a radius R2, equal to the segment AR, divide the circle into five equal parts. The starting point is chosen depending on the location of the pentagon. ( ! It is impossible to perform serifs in one direction, since errors occur and the last side of the pentagon turns out to be skewed.)

The division of a circle into 10 equal parts is performed similarly to the division of a circle into five equal parts ( rice. 28b), but first divide the circle into five parts, starting construction from point A, and then from point B, located at the opposite end of the diameter. Can be used to draw a segment OR- the length of which is equal to the chord 1/10 of the circumference.

Dividing the circle into 7 equal parts.

1/7

a B C)

Rice. 29

From anywhere (eg. BUT) circles, with a radius of a given circle, draw an arc until it intersects with a circle at points AT and D (Fig. 29, a). By connecting the dots AT and D straight, get a cut sun, equal to the chord that subtends an arc that is 1/7 of the circumference. Serifs are performed in the sequence indicated on rice. 29 b.

Pairings

Often in the design of parts, one surface passes into another. Usually these transitions are made smooth, which increases the strength of the parts and makes them more convenient to work with. Pairing is a smooth transition from one line to another. The construction of conjugations comes down to three points: 1) determining the center of conjugation; 2) finding junction points; 3) construction of an arc of conjugation of a given radius. To build a mate, the mate radius is most often specified. The center and junction point are defined graphically.

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 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 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).

Division of a circle into 3 equal parts.

To divide a circle of radius R into 3 equal parts and inscribe an equilateral triangle into it, from the point of intersection of the diameter with the circle (for example, from point A), an additional arc of radius R is described as from the center. Points 2 and 3 are obtained. Points 1, 2, 3 divide circle into three equal parts. By connecting straight lines points 1, 2, 3 build an inscribed equilateral triangle.

Division of a circle into 6 equal parts.

To divide the circle into 6 equal parts, two arcs of radius R are drawn from two opposite points (1 and 4) of the intersection of the diameter with the circle. Points (2, 3, 5, 6) are obtained. Together with the points that were obtained at the intersection of the diameter with the circle, he divides the circle into 6 equal parts.

Dividing a circle into 12 equal parts.

To divide the circle into 12 equal parts from the four points of intersection of the axes of symmetry with the circle, 4 arcs of radius R are described. The points obtained, together with those obtained by crossing the axes of symmetry with the circle, divide the circle into 12 equal parts.

Types of section designations in drawings

To show the transverse shape of parts, use images called sections (Fig. 13). In order to obtain a section, the part is mentally dissected by an imaginary cutting plane in the place where its shape needs to be revealed. The figure obtained as a result of cutting the part with a cutting plane is depicted in the drawing. Consequently a section is an image of a figure obtained by mentally dissecting an object by a plane or several planes.

The section shows only what is obtained directly in the cutting plane.

For clarity of the drawing, the sections are highlighted with hatching. Inclined parallel hatching lines are drawn at an angle of 45 ° to the lines of the drawing frame, and if they coincide in direction with the contour lines or center lines, then at an angle of 30 ° or 60 °.

Exposed section.

The contour of the rendered section is outlined with a solid thick line of the same thickness as the line adopted for the visible contour of the image. If the section is taken out, then, as a rule, an open line is drawn, two thickened strokes, and arrows indicating the direction of view. From the outside of the arrows, the same capital letters are applied. Above the section, the same letters are written through a dash with a thin line below. If the section is a symmetrical figure and is located on the continuation of the section line (dash-dotted line), then no designations are applied.

Superimposed section.

The contour of the superimposed section is a solid thin line (S/2 - S/3), and the contour of the view at the location of the superimposed section is not interrupted. The superimposed section is usually not indicated. But if the section is not a symmetrical figure, strokes of an open line and arrows are drawn, but letters are not applied.

Section designation

The position of the cutting plane is indicated in the drawing by a section line - an open line, which is drawn in the form of separate strokes that do not intersect the contour of the corresponding image. The thickness of the strokes is taken in the range from $ to 1 1/2 S, and their length is from 8 to 20 mm. On the initial and final strokes, perpendicular to them, at a distance of 2-3 mm from the end of the stroke, put arrows indicating the direction of view. At the beginning and end of the section line, they put the same capital letter of the Russian alphabet. The letters are applied near the arrows indicating the direction of view from the outside, fig. 12. Above the section, an inscription is made according to the type A-A. If the section is in a gap between parts of the same type, then with a symmetrical figure, the section line does not pass R4. The section can be rotated, then the symbol A-A must be added to the inscription

turned O, that is, A-AO.

When asked how to divide a circle into three equal parts with a compass)? tell me that please!! given by the author Embassy the best answer is
_______
Let a circle of radius R be given. We must divide it into three equal parts using a compass. Expand the compass by the radius of the circle. You can use a ruler for this, or you can put the compass needle in the center of the circle, and take the leg to the link describing the circle. In any case, the ruler will come in handy later.
Place the compass needle in an arbitrary place on the circle describing the circle, and with the stylus draw a small arc that intersects the outer contour of the circle. Then set the compass needle to the found reference point and once again draw an arc with the same radius (equal to the radius of the circle).
Repeat these steps until the next intersection point matches the very first one. You will get six reference circles spaced at regular intervals. It remains to select three points through one and connect them with a ruler to the center of the circle, and you will get a circle divided into three.
________
A circle can be divided into three parts if, using a compass, from the point of intersection of a straight line drawn through the center of the circle O, make the serifs B and C on the circle line with a compass equal to the radius of this circle.
Thus, two desired points will be found, and the third one is the opposite point A, where the circle and the line intersect.
Further, if necessary, with a ruler and pencil

you can draw an embedded triangle.

_________
For marking into three parts, use the radius of the circle.

Turn the compasses upside down. The needle is placed on
the intersection of the center line with the circle, and the stylus in the center. outline
an arc that intersects a circle.

The intersections will be the vertices of the triangle.