Show, how given a distance r, one can construct a regular pentagon, whose circumradius is r, using only ruler and compass...

Note 1: The ruler doesn't have any marks, so it's no good for measuring. It's only good for joining 2 points by a line.

Note 2: The circle which circumscribes the polygon, such that the polygon lies entirely within the circle and all of whose vertices lie on the circumference of the circle is the circumcircle of the polygon. The radius of this circle is the circumradius of the circle.

Of course, each angle in a regular polygon is [(5-2)180]/5=104°, not 360/5=72° as a few people suggested. An angle less than 90° doesn't make sense geometrically, of course, but I didn't even notice until just now.

A similarly weak method to constructing a 104° angle as my previous attempt, might be a right triangle with legs of x and 4x. One of the base angles is then invtan(4/1)=75.963°, which is supplementary to 104.037° (found by extending the shorter leg and using the external angle). Again, this is relatively close to 104°, but not nearly close enough for a credible solution.

I think that with insanely complex trig ratios, it would be possible to get very very close to the actual angle.
For example, tan 104°=--4.0107809335358447163457151294634..
To 5 places, that is -4.01078.
So, constructing a right angle, picking some arbitrary length, and adding that length 401078 times on one side and 10000 times on the other would give an external angle of
180°-invtan(401078/10000)
=180°-invtan(4.01078)
=180°-75.999996869569406544894796255462°
=-104.00000313043059345510520374454

Doing this with increased precision, while not entirely physically feasible, would produce an angle as good or better than any you would come up with using a regular protractor.

Again, this will never be exact, so, there must be another route to look down...

Posted by DJ on 2003-05-17 00:48:25