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Chapter 5 Dividing the Circle > 5.3 Constructing a Regular Pentagon - Pg. 98

98 CHAPTER 5 A F t B C O D E Figure 5.4. Constructing a regular pentagon. currently there are only five choices for the f j in theorem 5.1, although we can take any combination of these or none at all. This is really quite limiting because by telling us which n-gons are constructible, the theorem tells us which basic angles, or rather fractions of a circle, are not constructible. Hence, we have a very limited range of base angles from which to build our protractor. From a practical point of view, adding angles is not a sensible strategy, since errors are cumulative. More importantly, there is no way of obtaining an independent check of the scale. Hence, for our application the most important task is to divide angles. Once all angles have been marked out they can be checked against each other. 5.3 Constructing a Regular Pentagon To illustrate this excursion into polygons we shall explain the construction of a regular pentagon, shown in figure 5.4. Let us take a quadrant AOB in a unit circle, i.e. OA = 1 and the angle AOB is 90 . Mark C as the midpoint of the radius, then by the Pythagorean theorem 1 AC = 1 2 + ( 2 ) 2 = 5 2 .