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.

I have been contemplating this for some time and cannot think of any good way to even begin constructing this problem. If it is solved, then from that regular pentagon is now a way also to create a 72° angle. And perhaps, by similar methods if they exist, the angles formed by other regular polygons, such as 40°.

Perhaps the whole bit about r and the circumradius is overkill; simply constructing any regular pentagon or the required angle is going to be difficult, at least to come up with a method, and anything else is rather trivial.

Posted by DJ on 2003-05-16 23:59:05