Dividing a circle into three equal parts. Dividing a circle into any number of equal parts

Dividing a circle into six equal parts and constructing a regular inscribed hexagon is done using a square with angles of 30, 60 and 90º and/or a compass. When dividing a circle into six equal parts with a compass, arcs are drawn from two ends of the same diameter with a radius equal to the radius of the given circle until they intersect with the circle at points 2, 6 and 3, 5 (Fig. 2.24). By sequentially connecting the resulting points, 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 with a radius equal to the radius of the given circle is drawn until it intersects with the circle (Fig. 2.25). By connecting the resulting points, a dodecagon is obtained.

Figure 2.25

2.2.5 Dividing a circle into five and ten equal parts
and construction of 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, draw an arc 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 a chord subtending an arc whose length is equal to 1/5 of the circumference. Making notches on the circle (Fig. 2.26, in ) radius TO equal to segment 1B, divide the circle into five equal parts. Starting point 1 is chosen depending on the location of the pentagon. From point 1, build points 2 and 5 (Fig. 2.26, c), then from point 2, build point 3, and from point 5, build point 4. The distance from point 3 to point 4 is checked with a compass. If the distance between points 3 and 4 is equal to segment 1B, then the construction was carried out accurately. It is impossible to make serifs sequentially, in one direction, as errors occur and the last side of the pentagon turns out to be skewed. By sequentially connecting the found points, a pentagon is obtained (Fig. 2.26, d).

Dividing a circle into ten equal parts is carried out similarly to dividing a circle into five equal parts (Fig. 2.26), but first divide the circle into five parts, starting construction 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 a regular inscribed decagon (Fig. 2.27, b).

Figure 2.27

2.2.6 Dividing a circle into seven and fourteen equal parts
parts and construction of a regular inscribed heptagon and
quadragon

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

From any point on the circle, for example point A , draw an arc with the radius of a given circle (Fig. 2.28, a ) until it intersects with the circle at points B and D . Let's connect the points Vi D with a straight line. Half of the resulting segment (in this case segment BC) will be equal to the chord that subtends an arc constituting 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).

Dividing the circle into fourteen equal parts is done 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 sequentially by straight lines and a regular inscribed quadrangle is obtained (Fig. 2.29, b).

Figure 2.29

Construction of an ellipse

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

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

When constructing an ellipse depicting 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 ellipse axes (A, B, C, D), and the other four (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.

Sometimes, to make stencils, templates, drawings, patterns, and crafts, it is necessary to separate into 6 parts.
For example, we needed to make a template for a flower in the shape of a six-pointed star.

For those who have forgotten geometry, I remind you that you can divide a circle into 6 parts in two ways:

1. By using protractor.
2. By using compass.

1. How to divide a circle into 6 parts using a protractor

Dividing a circle using a protractor is very easy.

Draw a line connecting the center and any point (for example, point 1) on the circle. From this line, using a protractor, we plot an angle of 60, 120, 180 degrees. We put points on the circle (for example, points 2, 3, 4). We unfold the protractor and divide the other part of the circle in the same way.

2. How to divide a circle into 6 parts using a compass

It happens that you don’t have a protractor at hand. Then the circle can be divided into 6 equal parts using a compass.

Draw a circle, for example, with a radius of 5 cm (red circle). Without changing the radius, we move the leg of the compass to the circle (point 1) and draw another circle. We get two points of intersection of the black and red circles 6 and 2.

We move the leg of the compass to point 2 and again draw a circle. We get point 3.

We move the leg of the compass to point 3. Again we draw a circle.

Thus, we continue to divide the circle until we divide it into 6 equal parts.

A circle is the geometric locus of points on a plane that are equidistant from a given point, called the center, at a given non-zero distance, called its radius.

In this article you will learn how to divide a circle into 3-6, 4-8, 5-10 and n parts.

How to divide a circle into 3 and 6 parts

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

Dividing the circle into 3-6 equal parts

How to divide a circle into 5 and 10 parts

In order to divide a circle into 5 and 10 equal parts, it is necessary to construct a regular pentagon. To build it you need to do the following. We draw two mutually perpendicular circle axis equal to the diameter of the circle. Divide the right half of the horizontal diameter in half using arc R1. From the resulting point “a” in the middle of this segment with radius R2, draw a circular arc until it intersects with the horizontal diameter at point “b”. With radius R3, from point “1”, draw a circular arc until it intersects with a given circle (point 5) and obtain the side of a regular pentagon, then plot the resulting distance along the circle 5 times until a regular pentagon is obtained. The distance "b-0" gives the side of a regular pentagon.

Dividing the circle into 5-10 equal parts

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How to divide a circle into n equal parts

Otherwise, you need to construct a regular polygon with n number of sides. We draw horizontal and vertical mutually perpendicular axis of the circle. From the top point “1” of the circle, draw a straight line at an arbitrary angle to the vertical axis. On it we lay out 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 to the lower point of the vertical diameter. Draw a line parallel to the resulting one from the ends of the set aside segments until it intersects with the vertical diameter, thus dividing the vertical diameter of a given circle into a given number of parts. With a radius equal to the diameter of the circle, from the bottom 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 required ones, since points 1, 2,... 9 divide the circle into 9 (N) equal parts.

Dividing a circle into n equal parts

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The division of a circle into an arbitrary number of equal parts can be done using a table of chords, the numerical expression of which is determined by multiplying the radius of a given circle by the coefficient corresponding to the division number presented in the table.

Table of chords (coefficients for dividing a circle)

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How to find the center of a circular arc

It is necessary to do the following: on this arc we 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 using 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 its corresponding circle.

Approximate division of a circular arc into an arbitrary number of equal parts can be done using a compass using the method of successive approximation.

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

You will need

• - compass
• - ruler
• - protractor

Instructions

Let a circle of radius R be given. We need to divide it into three equal parts using a compass. Open the compass to the size of the radius of the circle. You can use a ruler, or you can place the needle of the compass in the center of the circle, and move the leg to the circle that describes the circle. In any case, the ruler will come in handy later. Place the compass needle in an arbitrary place on the circumference of the circle, and with a stylus, draw a small arc intersecting the outer contour of the circle. Then place the compass needle at the found intersection point and draw an arc again with the same radius (equal to the radius of the circle). Repeat these steps until the next intersection point coincides with the very first one. You will get six points on the circle, spaced at equal intervals. All that remains is to select three points through one and use a ruler to connect them 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 draw an angle of 120° three times on the outside of the circle and connect the resulting points on the circle with the center.

note

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

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

And construction of regular inscribed polygons

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

To construct a regular inscribed triangle, you need to start from a point A intersection of the center line with the circle, set aside a size equal to the radius R, one way or the other. We get vertices 1 and 2( rice. 26, a). Vertex 3 lies on the opposite point A end of the diameter.

1/3 1/6 1/12

a B C)

Rice. 26

The side of a 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, you need to place a size equal to the radius on the circle in one direction or the other from the four centers (Fig. 26, V).

Dividing a 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 a 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 point N, as if from the center, draw an arc with a radius R 1, equal to the distance from the point N to the point A, 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 equal to 1/5 of the circumference. Making notches 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. ( ! You cannot make serifs in one direction, as errors will accumulate and the last side of the pentagon will turn out skewed.)

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

Dividing a circle into 7 equal parts.

1/7

a B C)

Rice. 29

From any point (for example, A) circles with the radius of a given circle draw an arc until it intersects the circle at points IN And D (Fig. 29, a). Connecting the dots IN And D straight, get a segment sun, equal to the chord that subtends an arc constituting 1/7 of the circumference. Serifs are performed in the sequence indicated on rice. 29 b.

Mates

Often in the design of parts one surface merges into another. Usually these transitions are made smooth, which increases the strength of the parts and makes them more convenient to use. Pairing is a smooth transition from one line to another. The construction of mates comes down to three points: 1) determining the center of the mate; 2) finding connecting points; 3) construction of a conjugate arc of a given radius. To create a fillet, the fillet radius is most often specified. The center and mate point are determined graphically.