How to divide a circle into parts?

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In order to divide a segment or an angle intoequal parts, special skills are required. You just need to use a ruler or protractor. However, few know how to divide the circle into parts. Let's try to do it together.

Set for drawing

For work we need:

Ruler,
Compass,
A triangle with an angle of 90 degrees (optional).

Finding the center of a circle

Before proceeding to divide the circle into parts, it is necessary to find the center of this circle and to repulse in further constructions from it.

If the center of the circle is not initially assigned to us, we can find it ourselves.

Construct two arbitrary segments inside the circle, each of which connects two arbitrary arbitrary points on the circle. In other words, draw two chords.
Separate each of these segments with a ruler in half.
From the marked points of the middle of the segments, construct perpendiculars.
The intersection point of the perpendiculars will be the approximate point of the middle of the circle.

Division into two parts

After we found the center of a givencircle, we can not easily divide the circle into two equal parts. To do this, it is sufficient to draw a circle diameter that connects any two points of the circle and passes through its center.

Division into 3 equal parts

Divide the circle into three equal parts is also not as difficult as it seems at first glance. For this it is necessary.

Find the center of the circle, denoting it by the point O.
Construct the diameter of the circle MC.
From any obtained point of the end of the diameter, construct a new circle with a diameter equal to the diameter of the given circle.
Denote the intersection points of the circles by the letters A and B.
Carry out segments OA and OB.
The segments MO, OA and OB will divide the circle into three equal parts.

Division into 5 equal parts

Now we will dwell in greater detail on the process of dividing the circle into equal parts by the example of its division into 5 equal parts.

Find the center of the circle. Label it with a point O.
Draw the diameter of the circle: draw a segment that extends from any arbitrary point of the circle A that intersects the center and ends on the opposite side of the circle at point B. The point O will be the midpoint of this segment AB.
Construct a perpendicular to the segment AB at the point O. The most accurate construction will be if you alternately from points A and B draw circles with the same radii exceeding the length of the segment AO and OB, and then draw a straight line through the points of their intersection. It will be exactly perpendicular to the segment AB and pass through m. The resulting diameter is denoted by the letters M and M1.
In the same way, divide the segment AO and designate the resulting point with the letter C.
Next, using a compass, draw a circle centered at point C and with a radius equal to the segment CM.
Mark the intersection point of this circle with the segment AB with the letter K.
Mark the intersection points of this circle with the sides of the given initial circle by the letters T and X.
If we draw from the point M the segments passing through the points T, C, O, K, and X obtained by us, then we divide the given circle into 5 equal parts.

In a similar way, we can fit a regular pentagon into a given circle. To do this you need:

Repeat all steps from step 1 to step 7 of the previous algorithm.
Draw a circle from point T with a radius equal to the length of the segment CM. The intersection point of the resulting circle with this initially circle is denoted by the letter Y.
Draw a circle from X with a radius equal to the length of the segment CM. The point of intersection of the obtained circle with this initially circle is designated by the letter P.
Draw the segments UR, TU, TM, MX and XP. The resulting figure is the desired correct inscribed pentagon.

Division with protractor

If you need to quickly divide the circle into five equal parts, at the same time you have a protractor near your hand, and a small error in the construction does not frighten you, you can do the following:

Find the center of the circle.
Connect the center of the circle to any arbitrary point lying on the circle, i.e. draw the radius.
Put the protractor to the radius of the circle, set the angle to 72 degrees. The sector resulting from this division will be 1/5 of the circumference.

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