Divide the circle into 16 equal parts. Dividing a circle into equal parts

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DIVISION OF A CIRCLE INTO EQUAL PARTS

Some parts of machines and devices have elements evenly spaced around the circumference, for example, the parts in Fig. 52-59. When making drawings of such parts, you need to know the rules for dividing a circle into an equal number of parts.

Division of a circle into four and eight equal parts. On fig. 52, a shows a cover that has eight holes evenly spaced around the circumference. When constructing a drawing of the contour of the cover (Fig. 52 G) Divide the circle into eight equal parts. This can be done using a square with angles of 45 ° (Fig. 52, c), the hypotenuse of the square must pass through the center of the circle, or by construction.

Two mutually perpendicular diameters of a circle divide it into four equal parts (points 7, 3, 5, 7 in Fig. 52, b). To divide a circle into eight equal parts, the well-known division technique is used. right angle using a compass into two equal parts. Get points 2, 4, 6, 8.

Division of a circle into three, six and twelve equal parts. In the flange (Fig. 53, a) there are three holes evenly spaced around the circumference. When drawing the contour of the flange (Fig. 53, d), it is necessary to divide the circle into three equal parts.

To find points dividing a circle of radius R into three equal parts, enough from any point on the circle, for example, a point BUT, draw an arc with a radius R . The intersections of the arc with the circle give the two desired points 2 and 3; the third point of division will be located at the intersection of the axis of the circle drawn from the point L with the circle (Fig. 53, b).

You can also divide the circle into three equal parts with a square with angles of 30 and 60 ° (Fig. 53, c), the hypotenuse of the square must pass through the center of the circle.

On fig. 54, b shows the division of the circle by a compass into six equal parts. In this case, the same construction is performed as in Fig. 53, b but the arc is described not once, but twice, from points and radius R equal to the radius of the circle.

You can also divide the circle into six equal parts with a square with angles of 30 and 60 ° (Fig. 54, c). On fig. 54, a a cover is shown, when drawing which it is necessary to divide the circle into six parts.

To make a drawing of a part (Fig. 55, a), which has 12 holes evenly spaced along the circles, you need to divide the axial circle into 12 equal parts (Fig. 55, d).

When dividing a circle into 12 equal parts using a compass, you can use the same technique as when dividing a circle into six equal parts (Fig. 54, b), but arcs with a radius R describe four times from points 1, 7, 4 and 10 (Fig. 55, b).

Using a square with angles of 30 and 60 °, followed by turning it by 180 °, divide the circle into 12 equal parts (Fig. 55, in).

Division of a circle into five, ten and seven equal parts. In the die (Fig. 56, a) there are five holes evenly spaced around the circumference. When drawing a die (Fig. 56, c), it is necessary to divide the circle into five equal parts. Through the intended center O (Fig. 56, b)

with the help of a T-square and a square, axial lines are drawn and from point O they describe a circle of a given diameter with a compass. From point A with radius R equal to the radius of the given circle, an arc is drawn that intersects the circle at point n. From point n, a perpendicular is lowered to the horizontal center line, point C is obtained. From point C with radius R 1 equal to the distance from point C to point 1, an arc is drawn that intersects the horizontal center line at point t. From point 1 with radius R equal to the distance from point 1 to point m, draw an arc that intersects the circle at point 2. Arc 12 is 1/5 of the circumference. Points 3,4 and 5 are found by setting aside segments equal to m1 with a compass.

Detail "asterisk" (Fig. 57, a) has 10 identical elements evenly spaced around the circumference. To draw an asterisk (Fig. 57, i), the circle should be divided into 10 equal parts. In this case, the same construction should be applied as when dividing the circle into five parts (see Fig. 56, b). Line segment p 1 will be equal to the chord that divides the circle into 10 equal parts.

On fig. 58, a a pulley is shown, and in fig. 58, in- a drawing of a pulley, where the circle is divided into seven equal parts.

The division of the circle into seven equal parts is shown in Fig. 58b. From a point BUT an auxiliary arc is drawn with a radius R, equal to the radius of the given circle that intersects the circle at a point. From a point n lower the perpendicular to the horizontal center line. From a point 1 radius equal to the segment nс, make seven serifs around the circumference and get seven desired points.

Division of a circle into any number of equal parts. With sufficient accuracy, you can divide the circle into any number of equal parts, using the table of coefficients for calculating the length of the chord (Table 9).

Knowing how many (n) it is necessary to divide the circle, find the coefficient from the table. When multiplying the coefficient k by the diameter of the circle D, the length of the chord l is obtained, which is plotted with a compass on the circle n once.

When constructing a drawing of a ring (Fig. 59, a) it is necessary to divide a circle with a diameter of D \u003d 142 mm into 32 equal parts. The number of parts of the circle n=32 corresponds to the coefficient k=0.098. Calculate the length of the chord l= Dk= 142x0.098 \u003d 13.9 mm, it is laid with a compass on a circle 32 times (Fig. 59, b and in).

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 outside shooters apply the same capital letters. 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. An inscription is made above the section type A-A. If the section is in a gap between parts of the same type, then when symmetrical figure section line without passing R4. The section can be positioned with a rotation, then to inscriptions A-A symbol must be added

turned O, that is, A-AO.

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

Dividing a circle into equal parts, building regular polygons

Dividing a circle into 4 and 8 equal parts

Ends of mutually perpendicular diametersACandBD(Fig. 1) divide the circle centered at the pointOinto 4 equal parts. By connecting the ends of these diameters, you can get a squareAsunD.

If the angleSOAbetween mutually perpendicular diametersAEandFROMG(Fig. 2) divide in half and draw mutually perpendicular diametersD.H.andbf, then their ends will divide the circle centered at the pointOinto 8 equal parts. By connecting the ends of these diameters, you can get a regular octagonABCDEFGH.

Rice. 1 Fig. 2

Division of a circle into 3, 6 and 12 parts

To divide a circle into 6 equal parts, use the equality of the sides of a regular hexagon to the radius of the circumscribed circle. Given a circle centered at a pointO(Fig. 3) and radiusR, then from the ends of one of its diameters (pointsBUTandD), as from the centers, draw arcs of circles with a radiusR. The intersection points of these arcs with a given circle will divide it into 6 equal parts. Consistently connecting the found points, get the correct hexagonABCDEF.

If the circle is in the center with a dotO(Fig. 4) must be divided into 3 equal parts, then with a radius equal to the radius of this circle, an arc should be drawn from only one end of the diameter, for example, a pointD. pointsATandFROMintersection of this arc with a given circle, as well as a pointBUTdivide the latter into 3 equal parts. By connecting the dotsBUT, ATandFROM, you can get an equilateral triangleABC.

Rice. 3 Fig. four

To divide the circle into 12 parts, the division of the circle into 6 parts is repeated twice (Fig. 5), using the ends of mutually perpendicular diameters as centers: pointsBUTandG, DandJ. The intersection points of the drawn arcs with a given circle will divide it into 12 parts. By connecting the constructed points, you can get the correct dodecagon.

Rice. 5

Division of a circle into 5 parts

O(Fig. 6) into 5 parts, proceed as follows. One of the circle's radii, for exampleOM, divided in half by the previously described method. From the middle of the segmentOMdotNradiusR1 , equal to the segmentBUTN, draw an arc of a circle and mark a pointRintersection of this arc with the diameter to which the radius belongsOM. Line segmentARequal to the side of a regular pentagon inscribed in a circle. So from the endBUTdiameter perpendicular toOM, radiusR2 , equal to the segmentAR, draw an arc of a circle. pointsATandEintersections of this arc with a given circle make it possible to mark two vertices of the pentagon.

Two more topsFROMandD) are the points of intersection of arcs of circles with radiusR2 centered at pointsATandEwith a given circle centered at pointsO. Vertices of a regular pentagonABCDEdivide the given circle into 5 equal parts.

Rice. 6

Division of a circle into 7 parts

To divide a circle centered at a pointO(Fig. 6) into 7 parts, it is necessary to draw an auxiliary arc from point 1 with a radiusR, equal to the radius of the given circle, which intersects the circle at the pointM. From a pointNI lower the perpendicular to the horizontal center line. From a pointBUTwith a radius equal to the radiusMN, make 7 serifs around the circle and get seven desired points, connecting which get a regular heptagonABCDEFG.

Rice. 7

Dividing a circle into an arbitrary number of equal parts

If none of the options considered earlier satisfies the condition of the task, then a technique is used that allows you to divide the circle into an arbitrary number of equal parts and construct the corresponding inscribed in it regular polygons with an arbitrary number of sides.

Consider such a construction using the example of dividing a circle centered at a pointO(Fig. 8a) into 7 equal parts. First, you need to draw two mutually perpendicular diameters, one of which, for example, passing through a pointBUT, should be divided into 7 equal parts, limited by points 1 ... 7. From a pointBUT, as from the center, radiusRequal to the diameter of a given circle, it is necessary to draw an arc, the intersection of which with the continuation of the second diameter will determine the pointsR1 andR2 . Then through the dotsR1 andR2 (Fig. 8b), and even points obtained by dividing the diameterA7(points 2. 4 and 6), draw straight lines. pointsAT, FROM, DandE, F, Gintersection of these lines with a given circle and a pointBUTshare the circle with the centerOinto 7 equal parts. Consistently connecting the constructed points, you can draw a regular heptagon inscribed in a circle.

Rice. eight

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.