Dividing a circle into equal parts using a compass and straightedge. Dividing a circle into equal parts (how to divide)

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 circle 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, as 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.

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 image splitting effect 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!

Master class: Embroider a peacock

My first debut Master Class. Hopefully not the last. We will embroider a peacock. Product diagram.When marking the places of punctures, pay special attention so that they are in closed contours 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. How to transfer the scheme to the base. Embroidery consists of three layers thread. 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 ...

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 in this case, 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.

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 resulting points 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. By connecting all the 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 the length of the side of the 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

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 the point "1" draw an arc of a circle to the intersection with a given circle (p. 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.