Divide the circle into 3 equal parts. Dividing a circle into six equal parts and constructing a regular inscribed hexagon

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
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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.
Set the compass needle at 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.
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The 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 notches 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.

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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.

Division of a circle by six equal parts and the construction of a regular inscribed hexagon is performed using a square with angles of 30, 60 and 90 º and / or a compass. When dividing the circle into six equal parts with 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 (Fig. 2.24). Consistently connecting the points obtained, a regular inscribed hexagon is obtained.

Figure 2.24

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. 2.25). By connecting the points obtained, a dodecagon is obtained.

Figure 2.25

2.2.5 Division of a circle into five and ten equal parts
and construction of a regular inscribed pentagon and decagon

The division of a circle into five and ten equal parts and the construction of a regular inscribed pentagon and decagon is shown in Fig. 2.26.

Figure 2.26

Half of any diameter (radius) is divided in half (Fig. 2.26 a), 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 to the intersection with the second half of this diameter, at point B ( Fig. 2.26 b ). Segment 1 is equal to the chord that subtends the arc, the length of which is equal to 1/5 of the circumference. Making serifs on a circle (Fig. 2.26, in ) radius To, equal to the segment 1B, divide the circle into five equal parts. The starting point 1 is chosen depending on the location of the pentagon. Points 2 and 5 are built from point 1 (Fig. 2.26, c), 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 errors occur and the last side of the pentagon turns out to be skewed. Consistently connecting the points found, a pentagon is obtained (Fig. 2.26, d).

The division of the circle into ten equal parts is performed similarly to the division of the circle into five equal parts (Fig. 2.26), but first the circle is divided into five parts, starting from point 1, and then from point 6, located at the opposite end of the diameter (Fig. 2.27, a). By connecting all the points in series, they get the correct inscribed decagon (Fig. 2.27, b).

Figure 2.27

2.2.6 Division of a circle into seven and fourteen equal
parts and the construction of a regular inscribed heptagon and
tetradecagon

The division of a circle into seven and fourteen equal parts and the construction of a regular inscribed heptagon and a fourteen-gon is shown in Fig. 2.28 and 2.29.

From any point on the circle, for example point A , an arc is drawn with a radius of a given circle (Fig. 2.28, a ) to the intersection with the circle at points B and D . Connect the points B and D with 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, notches are made on the circle in the sequence shown in Fig. 2.28, b . By connecting all the points in series, they get a regular inscribed heptagon (Fig. 2.28, c).

The division of the circle into fourteen equal parts is performed by dividing the circle into 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 performed from point 8 . The constructed points are connected in series with straight lines and get a regular inscribed fourteen (Fig. 2.29, b).

Figure 2.29

Building an ellipse

Image of a circle in a rectangular isometric view in all three projection planes is an ellipse of the same shape.

The direction of the minor axis of the ellipse coincides with the direction of the axonometric axis, perpendicular to the plane of projections in which the depicted circle lies.

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

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.

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.

When performing graphic work, you have to solve many construction tasks. The most common tasks in this case are the division of line segments, angles and circles into equal parts, the construction of various conjugations.

Dividing a circle into equal parts using a compass

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

Division of a circle into four equal parts.

Dash-dotted center lines drawn perpendicular to one another divide the circle into four equal parts. Consistently connecting their ends, we get a regular quadrilateral(Fig. 1) .

Fig.1 Division of a circle into 4 equal parts.

Division of a circle into eight equal parts.

To divide a circle into eight equal parts, arcs equal to the fourth part of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of the circle, notches are made outside it. The points obtained are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, i.e., eight equal sections of the circle are obtained (Fig. 2 ).

Fig.2. Division of a circle into 8 equal parts.

Division of a circle into sixteen equal parts.

Dividing an arc equal to 1/8 into two equal parts with a compass, we will put serifs on the circle. Connecting all serifs with straight line segments, we get a regular hexagon.

Fig.3. Division of a circle into 16 equal parts.

Division of a circle into three equal parts.

To divide a circle of radius R into 3 equal parts, from the point of intersection of the center line 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 the circle into three equal parts.

Rice. four. Division of a circle into 3 equal parts.

Division of a circle into six equal parts. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle (Fig. 5.).

To divide a circle into six equal parts, it is necessary from points 1 and 4 intersection of the center line with the circle, make two serifs on the circle with a radius R equal to the radius of the circle. Connecting the obtained points with line segments, we get a regular hexagon.

Rice. 5. Dividing the circle into 6 equal parts

Division of a circle into twelve equal parts.

To divide a circle into twelve equal parts, it is necessary to divide the circle into four parts with mutually perpendicular diameters. Taking the points of intersection of the diameters with the circle BUT , AT, FROM, D beyond the centers, four arcs are drawn by the radius to the intersection with the circle. Received points 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and points BUT , AT, FROM, D divide the circle into twelve equal parts (Fig. 6).

Rice. 6. Dividing the circle into 12 equal parts

Dividing a circle into five equal parts

From a point BUT draw an arc with the same radius as the radius of the circle before it intersects with the circle - we get a point AT. Lowering the perpendicular from this point - we get the point FROM.From point FROM- the midpoint of the radius of the circle, as from the center, by an arc of radius CD make a notch on the diameter, get a point E. Line segment DE equal to length sides of an inscribed regular pentagon. By making a radius DE serifs on the circle, we get the points of dividing the circle into five equal parts.

Rice. 7. Dividing the circle into 5 equal parts

Dividing a circle into ten equal parts

By dividing the circle into five equal parts, you can easily divide the circle into 10 equal parts. Having drawn straight lines from the resulting points through the center of the circle to the opposite sides of the circle, we get 5 more points.

Rice. 8. Dividing the circle into 10 equal parts

Dividing a circle into seven equal parts

To divide a circle of radius R into 7 equal parts, from the point of intersection of the center line with the circle (for example, from the point BUT) describe how from the center an additional arc the same radius R- get a point AT. Dropping a perpendicular from a point AT- get a point FROM.Line segment Sun equal to the length of the side of the inscribed regular heptagon.

Rice. 9. Dividing the circle into 7 equal parts

1. BRIEF THEORETICAL INFORMATION

1.1. Geometric constructions

Dividing a circle into equal parts

Some parts have elements evenly distributed around the circumference. When making drawings of parts with similar elements, it is necessary to be able to divide the circle into equal parts. Techniques for dividing a circle into equal parts are shown in fig. one

Rice. 1. Dividing the circle into equal parts

With sufficient accuracy, you can divide the circle into any number of equal parts using a table of coefficients to calculate the length of the stroke.

By the number of equal segments on the circle (table 1) we find the corresponding coefficient. When multiplying the obtained coefficient by the diameter of the circle, we get the length of the chord, which we put on the circle with a compass.

Table 1 - Coefficient for determining the length of the chord

Making a Pairing Between Two Lines

When drawing the contours of technical details and in other technical constructions, it is often necessary to perform conjugations (smooth transitions) from one line to another. The pairing of two sides of the angle with an arc given to the radius of the arc R is performed in the following sequence:

- parallel to the sides of the corner at a distance equal to R, two auxiliary straight lines are drawn;

- the point of intersection of these lines will be the center of conjugation;

- from the center of conjugation, perpendiculars are made to the given lines;

- the points of intersection of perpendiculars with given lines are called conjugation points;

- an arc with radius R is built from the center of the junction, connecting the junction points.

On fig. 2 shows examples of constructing mates when the radius of the mate arc is specified. In this case, it is necessary to define the mate center and mate points. The contour of the part is drawn using a compass.

Rice. 2. Techniques for constructing conjugations

In technology, it is often necessary to draw curved lines made up of a large number small arcs of circles with a gradual change in the radius of their curvature. Such lines cannot be drawn with a compass. These curves are drawn with the help of curves and are called patterns. It is necessary to study the regularity of the formation of a curved curve and put on the drawing a number of points belonging to it. The points are connected by a smooth curve with a thin freehand line, and the stroke is performed using a pattern.

To trace pattern curves, you need to have a set of several patterns. Having chosen a suitable template, the edge of the part of the template is adjusted to the largest possible number of found points. To circle

the next section, you need to adjust the edge of the pattern to two or three more points, while the pattern should touch a part of the already circled curve. The method of drawing a curve along the pattern is shown in fig. 3.

Rice. 3. Construction of a curve on a template.

On fig. 4 shows an example of constructing an ellipse along given axes

Rice. 4. Building an ellipse

On fig. Figure 5 shows an example of constructing a parabola by dividing the sides of the angle AOC into the same number of equal parts. On fig. 6 gives an example of constructing the involute of a circle. Set

The circle is divided into 12 equal parts. The tangents to the circle are drawn through the division points. On the tangent drawn through the point 12, the length of this circle is plotted and divided into 12 equal parts. Starting from the point l on the tangents to the circle, successively lay off segments equal to 1/12 of the circumference, 1/6, 1/4, etc.

Rice. 5. Construction of a parabola

Rice. 6. Construction of the involute

Rice. 7. Construction of a sinusoid

Fig. 8 Construction of the Archimedes spiral

On fig. 7 shows the technique for constructing a sinusoid. A given circle is divided into 12 equal parts, a straight line segment is divided into the same number of equal parts, equal to the length of the unfolded