During renovations, you often have to deal with circles, especially if you want to create interesting and original decorative elements. You also often have to divide them into equal parts. There are several methods to do this. For example, you can draw regular polygon or use tools known to everyone since school. So, in order to divide a circle into equal parts, you will need the circle itself with a clearly a certain center, pencil, protractor, as well as ruler and compass.
Dividing a circle using a protractor
Dividing a circle into equal parts using the above-mentioned tool is perhaps the simplest. It is known that a circle is 360 degrees. By dividing this value into the required number of parts, you can find out how much each part will take (see photo).
Next, starting from any point, you can make notes corresponding to the calculations performed. This method is good when you need to divide a circle by 5, 7, 9, etc. parts. For example, if the shape needs to be divided into 9 parts, the marks will be at 0, 40, 80, 120, 160, 200, 240, 280 and 320 degrees.
Division into 3 and 6 parts
To correctly divide a circle into 6 parts, you can use the property regular hexagon, i.e. its longest diagonal must be twice the length of its side. To begin with, the compass must be stretched to a length equal to the radius of the figure. Next, leaving one of the legs of the tool at any point on the circle, the second one needs to make a notch, after which, repeating the manipulations, you will be able to make six points, connecting which you can get a hexagon (see photo).
By connecting the vertices of the figure through one, you can get regular triangle, and accordingly the figure can be divided into 3 equal parts, and by connecting all the vertices and drawing diagonals through them, you can divide the figure into 6 parts.
Division into 4 and 8 parts
If the circle needs to be divided into 4 equal parts, first of all, you need to draw the diameter of the figure. This will allow you to get two of the required four points at once. Next, you need to take a compass, stretch its legs along the diameter, then leave one of them at one end of the diameter, and make the other notches outside the circle from below and above (see photo).
The same must be done for the other end of the diameter. After this, the points obtained outside the circle are connected using a ruler and pencil. The resulting line will be a second diameter, which will run clearly perpendicular to the first, as a result of which the figure will be divided into 4 parts. In order to get, for example, 8 equal parts, the resulting right angles can be divided in half and diagonals drawn through them.
Dividing a circle into four equal parts and constructing a regular inscribed quadrilateral(Fig. 6).
Two mutually perpendicular center lines divide the circle into four equal parts. By connecting the points of intersection of these lines with the circle with straight lines, a regular inscribed quadrilateral is obtained.
Dividing a circle into eight equal parts and constructing a regular inscribed octagon(Fig. 7).
Dividing a circle into eight equal parts is done using a compass. in the following way.
From points 1 and 3 (points of intersection of the center lines with the circle) arbitrary radius R draw arcs until they intersect each other, and with the same radius from point 5 make a notch on the arc drawn from point 3.
Straight lines are drawn through the intersection points of the serifs and the center of the circle until they intersect with the circle at points 2, 4, 6, 8.
If the resulting eight points are connected sequentially by straight lines, you will get a regular inscribed octagon.
Dividing a circle into three equal parts and constructing a regular inscribed triangle(Fig. 8).
Option 1.
When dividing a circle with a compass into three equal parts, from any point on the circle, for example, point A of the intersection of the center lines with the circle, draw an arc of radius R, equal to the radius circles, we get points 2 and 3. The third division point (point 1) will be located at the opposite end of the diameter passing through point A. By connecting points 1, 2 and 3 in series, we get a regular inscribed triangle.
Option 2.
When constructing a regular inscribed triangle, if one of its vertices is given, for example point 1, find point A. To do this, through given point carry out the diameter (Fig. 8). Point A will be located at the opposite end of this diameter. Then an arc of radius R equal to the radius of the given circle is drawn, points 2 and 3 are obtained.
Dividing a circle into six equal parts and constructing a regular inscribed hexagon(Fig.9).
When dividing a circle into six equal parts using 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. By sequentially connecting the resulting points, a regular inscribed hexagon is obtained.
Dividing a circle into twelve equal parts and constructing a regular inscribed dodecagon(Fig. 10).
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. 10). By connecting sequentially obtained intersection points, a regular inscribed dodecagon is obtained.
Dividing a circle into five equal parts and constructing a regular inscribed pentagon ( Fig. 11).
When dividing a circle with a compass, half of any diameter (radius) is divided in half to obtain point A. 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. Segment 1B equal to chord subtending an arc whose length is equal to 1/5 of the circumference. Making notches on a circle of radius R1, equal to the segment 1B, divide the circle into five equal parts. The starting point A is chosen depending on the location of the pentagon.
From point 1, construct points 2 and 5, then from point 2, construct point 3, and from point 5, construct 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 notches sequentially, in one direction, since measurement errors accumulate and the last side of the pentagon turns out to be skewed. By sequentially connecting the found points, a regular inscribed pentagon is obtained.
Dividing a circle into ten equal parts and constructing a regular inscribed decagon(Fig. 12).
Dividing a circle into ten equal parts is carried out similarly to dividing a circle into five equal parts (Fig. 11), but first divide the circle into five equal parts, starting construction from point 1, and then from point 6, located at the opposite end of the diameter. By connecting all the points in series, a regular inscribed decagon is obtained.
Dividing a circle into seven equal parts and constructing a regular inscribed heptagon(Fig. 13).
From any point on a circle, for example point A, an arc is drawn with the radius of a given circle until it intersects with the circle at points B and D of the straight line.
Half of the resulting segment (in 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 when constructing a regular pentagon. By connecting all the points in sequence, a regular inscribed heptagon is obtained.
Dividing a circle into fourteen equal parts and constructing a regular inscribed quadrangle (Fig. 14).
Dividing a circle into fourteen equal parts is carried out similarly to dividing a circle into seven equal parts (Fig. 13), but first divide the circle into seven equal parts, starting construction from point 1, and then from point 8, located at the opposite end of the diameter. By connecting all the points in series, a regular inscribed quadrangle is obtained.
To divide a circle in half, it is enough to draw any diameter. Two mutually perpendicular diameters will divide the circle into four equal parts (Figure 28, a). By dividing each fourth part in half, you get eighth parts, and with further division - sixteenth, thirty-second parts, etc. (Figure 28, b). If connect the division points with straight lines, then you can get the sides of a regular inscribed square( a 4 ), octagon ( a 8 ) and t . d. (Figure 28, c).
Figure 28
Dividing a circle into 3, 6, 12, etc. equal parts, and construction of corresponding regular inscribed polygons carried out as follows. Two mutually perpendicular diameters are drawn in a circle 1–2 And 3–4 (Figure 29 a). From points 1 And 2 how arcs with the radius of a circle are described from centers R before intersecting it at points A, B, C And D . Points A , B , 1, C, D And 2 divide the circle into six equal parts. These same points, taken through one, will divide the circle into three equal parts (Figure 29, b). To divide a circle into 12 equal parts, describe two more arcs with the radius of the circle from points 3 And 4 (Figure 29, c).
Figure 29
You can also construct regular inscribed triangles, hexagons, etc. using a ruler and a 30 and 60° square. Figure 30 shows a similar construction for an inscribed triangle.
Figure 30
Dividing a circle into seven equal parts and the construction of a regular inscribed heptagon (Figure 31) is performed using half the side of the inscribed triangle, approximately equal side inscribed heptagon.
Figure 31
To divide a circle into five or ten equal parts draw two mutually perpendicular diameters (Figure 32, a). Radius O.A. divide in half and, having received a point IN , describe an arc from it with a radius R=BC until it intersects at the point D with horizontal diameter. Distance between points C And D equal to the side length of a regular inscribed pentagon ( a 5 ), and the segment O.D. equal to the length of the side of a regular inscribed decagon ( a 10 ). Dividing a circle into five and ten equal parts, as well as constructing inscribed regular pentagon and decagon are shown in Figure 32, b. An example of the use of dividing a circle into five parts is a five-pointed star (Figure 32, c).
Figure 32
Figure 33 shows general method approximate division of a circle into equal parts . Suppose you want to divide a circle into nine equal parts. Two mutually perpendicular diameters and a vertical diameter are drawn in a circle AB divided into nine equal parts using an auxiliary straight line (Figure 33, a). From the point B describe an arc with radius R = AB, and at its intersection with the continuation of the horizontal diameter, points are obtained WITH And D . From points C And D through even or odd diameter division points AB conduct rays. The intersection points of the rays with the circle will divide it into nine equal parts (Figure 33, b).