Using a compass and a ruler is there a way to divide a circle into equal parts 5 6 7 8 or more parts?

Answer 1

I can answer for 6, 12, 24 48, etc, 8, 16, 32, 64, etc with a slight "cheat"

First:

• To construct the perpendicular bisector of a line (with known end points):

1. Set the compass to larger than half the length of the line.
2. With the compass on one end point, draw arcs either side of the line
3. With the compass on the other end point, draw arcs to intersect these.
4. With the straight edge draw a line between the points where the arcs intersect on either side of the original line - this is the perpendicular bisector.

• To construct the bisector of an angle.

1. Set the compass to any width.
2. With the compass on the point of the angle, draw arcs on each arm of the angle.
3. With the compass on one arm of the angle where the arc intersects it, draw another arc between the arms (away from the point of the angle)
4. With the compass on the other intersection of arm and arc, draw another arc to intersect the arc just drawn.
5. Join the point of the angle to the point the two arcs intersect - this is the angle bisector.

In all cases, the centre of the circle is needed. If the centre is not known it can be found:

1. Draw any two chords on the circle.
2. Construct the perpendicular bisectors of the chords
3. Where the perpendicular bisectors meet is the centre of the circle.

• 6, 12, 24, 48, ...

For 6 equal parts:

1. If the circle has just been drawn, keep the compass set to its radius; otherwise set the compass to the radius of the compass (this is the slight "cheat") using the centre of the circle.
2. Mark the circumference of the circle at any point.
3. With the compass on this point, draw an arc to intersect the circle.
4. Using this intersection as the marked point, repeat step 3 three more times.
5. Join each of the six marked points on the circumference to the centre - the circle is now split into 6 equal parts.

For 12 equal parts:

1. Split the circle into 6 equal parts.
2. Bisect each of the segments (at the angle at the centre of the circle), joining the circle centre to the circumference each time.

For 24, 48, etc parts, bisect the segments of the previous one.

For example, to get 48 equal parts split the circle into 6 equal parts, bisect the angles to get 12 equal parts, bisect the angles again to get 24 equal parts and finally bisect the angles again to get 48 equal parts.

• 8, 16, 32, 64, ...

For 8 equal parts:

1. Draw in a diameter (2 equal parts).
2. Construct the perpendicular bisector of this diameter (4 equal parts).
3. Bisect each of the segments (at the centre angle), joining the centre to the circumference.

For 16, 32, 64, etc equal parts, bisect the segments (as with 12, 24, 48, etc parts).

In these constructions, there are an even number of points and the opposite points are at the ends of a diameter, so construct time can be reduced by drawing in diameters instead of radii.