Activity 6
Constructions

Any regular polygon can be drawn using a straightedge, compass, and protractor. In ancient Greece, it was the custom to attempt to draw regular polygons using only a straightedge and compass. Since the Mira is equivalent to a compass for many of its functions, we shall also include the Mira. Gauss found that any regular polygon of n sides can be constructed with an unmarked ruler and a compass if all of the following conditions are met:

Each odd factor of n is unique.
Each odd factor of n is prime.
Each odd factor of n is of the form
22 k + 1, for some integer k

This expression takes on the following values for different substitutions for k.

k 0 1 2 3 4 5
22 k + 1 3 5 17 257 65537 4294967297

Of the six numbers shown, all but the last are prime. Therefore, the numbers 3, 5, 17, 257, and 65537 can all be factors of n, where n is the number of sides of a regular polygon. However, there can be no other odd factors of n.

It would be possible to construct a regular 60-sided polygon with a compass and a straightedge, since the only odd factors of 60 are 3 and 5, and each different odd factor is unique. On the other hand, it is impossible to construct a regular 100-sided polygon with a compass and a straightedge, since 100 has more than one factor of 5. A 21-sided regular polygon cannot be constructed with compass and straightedge because 21 has an odd factor of 7, which is not in the table above.

The values of n less than 100 for which a regular n-sided polygon can be constructed using only a compass and straightedge are listed below:

3 4 5 6 8 10 12 15 16 17 20 24
30 32 34 40 48 51 60 64 68 80 85 96

If a regular polygon has the number of its sides other than in the above list, it can be constructed by using a protractor in addition to the compass and straightedge. In each case, a circle is drawn, the circle is divided into a number of arcs equal to the number of sides of the regular polygon, and the endpoints of the arcs are connected together with segments. To construct a regular nonagon (9 sides) mark off 40 degrees of arc around the circle for each side and connect the marks.

Regular Hexagon and Equilateral Triangle

The regular hexagon and equilateral triangle are drawn using the same technique. First, draw a circle of a given radius. Second, using the same radius to mark off arcs around the circumference of the circle. Connect consecutive marks to form a regular hexagon. Connect every second mark to form an equilateral triangle.

Regular Octagon and Square

The regular octagon and square are drawn using the same technique. First, draw a circle of a given radius. Second, draw any diameter of the circle. Third, use a Mira to draw another diameter that is perpendicular to the first diameter. Finally, bisect each of the right angles formed by the intersection of the diameters. Connect the endpoints of all four diameters in order around the circle to form the regular octagon. Connect every second endpoint to form a square.

Regular Pentagon

The regular pentagon can be drawn by following the steps below:

  1. Draw a circle with center C.
  2. Draw a diameter AB.
  3. Use a Mira to construct a perpendicular to AB that intersects circle at P.
  4. Use Mira to find midpoint of BC and label it D.
  5. Place point of compass at D and open pencil end to meet point P.
  6. Draw arc from P to diameter AB, meeting AB at point E.
  7. Place point of compass at P and open pencil end to meet point E.
  8. Use this compass setting to mark off arcs around the circle.
  9. Connect marks on circle to form regular pentagon.

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